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Cellular automata in operational probabilistic theories

Paolo Perinotti

QUIT Group, Dipartimento di Fisica, Università degli studi di Pavia, and INFN sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

The theory of cellular automata in oper-

ational probabilistic theories is developed.

We start introducing the composition of

inﬁnitely many elementary systems, and

then use this notion to deﬁne update rules

for such inﬁnite composite systems. The

notion of causal inﬂuence is introduced,

and its relation with the usual property of

signalling is discussed. We then introduce

homogeneity, namely the property of an

update rule to evolve every system in the

same way, and prove that systems evolving

by a homogeneous rule always correspond

to vertices of a Cayley graph. Next, we de-

ﬁne the notion of locality for update rules.

Cellular automata are then deﬁned as ho-

mogeneous and local update rules. Finally,

we prove a general version of the wrapping

lemma, that connects CA on diﬀerent Cay-

ley graphs sharing some small-scale struc-

ture of neighbourhoods.

Contents

1 Introduction 1

2 Detailed outlook 3

3 Operational Probabilistic Theories 4

3.1 Formal framework . . . . . . . . . 5

3.2 Causality and the no-restriction

hypothesis . . . . . . . . . . . . . . 8

3.3 Norms . . . . . . . . . . . . . . . . 10

4 The quasi-local algebra in OPTs 13

4.1 Quasi-local eﬀects . . . . . . . . . . 13

4.2 Extended states . . . . . . . . . . . 19

4.3 Quasi-local transformations . . . . 22

5 Global update rules 30

5.1 Update rule . . . . . . . . . . . . . 31

5.2 Admissibility and local action . . . 34

Paolo Perinotti: paolo.perinotti@unipv.it,

http://www.qubit.it

5.3 Causal inﬂuence . . . . . . . . . . . 36

5.3.1 Relation with signalling . . 38

5.4 Block decomposition . . . . . . . . 38

6 Homogeneity 40

7 Locality 47

8 Cellular Automata 50

8.1 Results . . . . . . . . . . . . . . . . 51

9 Examples 54

9.1 Classical case . . . . . . . . . . . . 54

9.2 Quantum case . . . . . . . . . . . . 55

9.3 Fermionic case . . . . . . . . . . . 55

10 Conclusion 56

Acknowledgments 56

A Identiﬁcation of the sup- and opera-

tional norm for eﬀects 56

B Quasi-local states 57

C Proof of identity 94 63

D Homogeneity and causal inﬂuence 64

E Proof of right and left invertibility of

a locally deﬁned GUR 64

F Proof of theorem 10 65

G Proof of lemmas 66 and 67 66

References 67

1 Introduction

In the last two decades a new approach to quan-

tum foundations arose, grounded on the quan-

tum information experience [1, 2, 3, 4]. This line

of research on the fundamental aspects of quan-

tum theory beneﬁts from new ideas, concepts and

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arXiv:1911.11216v3 [quant-ph] 6 Jul 2020

methods [5, 6, 7] that lead to a wealth of remark-

able results [8, 9, 10, 11]. In particular, quan-

tum theory can be now understood as a theory

of information processing, that is selected among

a universe of alternate theories [12, 13, 1, 14] by

operational principles about the possibility or im-

possibility to perform speciﬁc information pro-

cessing tasks [9, 15].

The scenario of alternate theories among which

the principles select quantum theory is called the

framework of Operational Probabilistic Theories

(OPTs) [8, 15], and has connections with the less

structured concept of Generalized Probabilistic

Theory (GPT) [5, 16, 17], as well as the dia-

grammatic category theoretical approach often

referred to as quantum picturialism [18, 19]. In-

spiration for the framework came also from quan-

tum logic [20, 21].

Quantum theory, namely the theory of Hilbert

spaces, density matrices, completely positive

maps and POVMs (in particular we refer to the

elegant exposition of Ref. [22]), can thus be re-

formulated as a special theory of information pro-

cessing. Besides the many advantages of this re-

sult, one has to face a main issue: the theory

as such is devoid of its physical content. El-

ementary systems are thought of as elementary

information carriers—brutally speaking, memory

cells—rather than elementary particles or ﬁelds in

space-time. While this framework is satisfactory

for an eﬀective, empirical description of physical

experiments, when it comes to provide a theo-

retical foundation for the physics of elementary

systems, the informational approach at this stage

calls for a way to re-embrace mechanical notions

such as mass, energy, position, space-time, and

complete the picture encompassing the dynamics

of quantum systems.

A recent proposal for this endeavour is based

on the idea that physical laws have to be ulti-

mately understood as algorithms, that make sys-

tems evolve, changing their state, exactly as the

memory cells of a computer are updated by the

run of an algorithm [23, 24, 25]. Such a program

already achieved successful results in the recon-

struction of Weyl’s, Dirac’s and Maxwell’s equa-

tions (for a comprehensive review see Ref. [26]).

The most natural candidate algorithm for de-

scribing a physical law in this context is a cellular

automaton. The theory of cellular automata is

a wide and established branch of computer sci-

ence. The notion of a cellular automaton for

quantum systems was ﬁrst devised as the quan-

tum version of its classical counterpart, e.g. in

Refs. [27, 28, 29, 30], but turned out to give rise to

a rather independent theory, developed starting

from Ref. [31]: the theory of Quantum Cellular

Automata (QCAs). The latter counts presently

various important results—see e.g. Refs. [32, 33],

just to mention a few. We stress that, most

commonly, QCAs are deﬁned to be reversible al-

gorithms, and most results in the literature are

proved with this hypothesis. It is known, how-

ever, that many desirable preoperties fail to hold

in the irreversible case (see e.g. Ref. [34]). In the

present work we will not consider irreversible cel-

lular automata, and leave this subject for further

studies.

In order to use cellular automata as candi-

date physical laws in the foundational perspec-

tive based on OPTs, one has two choices at

hand. The ﬁrst one is to start treating automata

within a deﬁnite theory, and this is the approach

adopted so far, in particular within Fermionic

theory [35, 36, 37]. In the linear case, Fermionic

cellular automata reduce to Quantum Walks, and

this brings in the picture all the tools from such

widely studied topic [38, 39, 40, 41, 42, 43]. The

non-linear case is far less studied [44, 45], and

does not oﬀer as many results for the analy-

sis. Needless to say, this approach faces diﬃcul-

ties that are speciﬁc of the theory at hand, and

prevents a comparison of diﬀerent theories on a

ground that is genuinely physical.

The second approach is initiated in the present

paper, and consists in deﬁning cellular automata

in the general context of OPTs. This perspective

oﬀers the possibility of extracting the essential

features of the theory of quantum or Fermionic

cellular automata, those that are not speciﬁc of

the theory but are well suited in any theory of

information processing. As a consequence, one

can generalise some results, and ﬁgure out why

and how others fail to extend to the broader sce-

nario. Moving a much less structured mathemat-

ical context, this approach can use only few tools,

but provides results that have the widest appli-

cability range.

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2 Detailed outlook

In this section we provide a short, non technical

discussion of the main results. The purpose of

this work is to deﬁne cellular automata in OPTs,

and prove some general results that will help ap-

plying the theory to special cases of interest.

A cellular automaton is an algorithm that up-

dates the information stored in an array of mem-

ory cells in discrete steps, in such a way that one

needs to read the content of a few neighbouring

cells to determine the state of a given cell at the

next step. Cellular automata in the literature

are often, but not always, deﬁned to be homoge-

neous: in this case the local rule for the update

of the cell is the same for every cell. Here we

will adopt homogeneity, but the subject is pre-

sented so that the generalisation of deﬁnitions to

inhomogeneous CA is straightforward.

Typically, the interesting case of a cellular au-

tomaton is the one involving an inﬁnite memory

array. The ﬁrst challenge we have to face is then

to extend the theory of OPTs from ﬁnite, arbi-

trarily large composite systems to actually inﬁ-

nite ones. This piece of theory has an interest

per se, for many reasons ranging from the possi-

bility to introduce thermodynamic limits to the

extension of the theory of C

∗

and von Neumann

algebras.

We then start with a review of the framework

of OPTs, and build the necessary tools to deﬁne

inﬁnite composite systems. The starting point

is the construction of mathematical objects that

describe measurements on ﬁnitely many systems

within an inﬁnite array— the OPT counterpart

of the space of local eﬀects of quantum theory.

Eﬀects for the inﬁnite system are then deﬁned as

limits of Cauchy sequences of local eﬀects. Since

the introduction of a suitable topology for the def-

inition of limits is needed, we open the paper with

section 3, where a review of OPTs is provided,

along with a few new results that will be useful,

and a rather consistent part of the section will be

dedicated to the introduction and discussion of

norms that will provide the necessary topological

framework for a consistent deﬁnition of limits.

The subject of the subsequent Section 4 is then

the construction of the Banach space of eﬀects for

the inﬁnite composite system, and consequently

the construction of the space of states as suitable

linear functionals on eﬀects. An important sub-

section will be dedicated to the construction of

the algebra of quasi-local transformations, that

allows for the description of transformations on

an inﬁnite system that can be arbitrarily well ap-

proximated by local operations on ﬁnitely many

subsystems. Indeed, an important part of the

theory of CAs in OPTs is built by ruling the way

in which quasi-local transformations are trans-

formed by the CA. In particular, the way in which

the CA propagates the eﬀects of a local transfor-

mation on surrounding subsystems will be the key

to the deﬁnition of the neighbourhood of the sub-

system, a concept that is central to the theory of

CAs.

Once this is done, the next step consists in

deﬁning update rules, and their admissibility con-

ditions, which are the subject of Section 5, along

with causal inﬂuence and a block-decomposition

theorem. Update rules are deﬁned in the ﬁrst

place as automorphisms V of the space of quasi-

local eﬀects, but an important request they must

abide is that when they act on a quasi-local trans-

formation A by conjugation as V A V

−1

, the ob-

tained transformation is again quasi-local.

The next step is taken in section 6, and con-

sists in deﬁning the property of homogeneity. The

latter presents with some diﬃculties, stemming

from the fact that, as we mentioned above, we

are deﬁning update rules prior to any mechanical

notion. This implies that even space-time is not

available at this fundamental stage, and without

geometry we cannot deﬁne homogeneity as trans-

lational invariance under the group correspond-

ing to a given space-time. On the contrary, we

will deﬁne homogeneity by formalising the idea

that every single cell has to be treated equally

by the update rule, and this notion will be de-

ﬁned operationally, requiring that no experiment

made of local operations will allow establishing

any diﬀerence between cells. As a consequence

of homogeneity, one can prove that every homo-

geneous update rule is underpinned by the Cay-

ley graph of some group, generalising a result of

Refs. [25, 46]. The mathematical theory known as

geometric group theory then tells us that the Cay-

ley graph representing the causal connections of

cells in the memory array, being a metric space,

uniquely identiﬁes an equivalence class of met-

ric spaces, that captures both the algebraic and

geometric essential features of the group. Aston-

ishingly, for Cayley graphs of ﬁnitely presented

groups—which is the case for cellular automata

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as we deﬁne them here—the equivalence class al-

ways contains a smooth manifold of dimension at

most four (see Ref. [47], pag. 90). This result

means that we can always think of a cellular au-

tomaton as if it was embedded in a Riemannian

manifold, in such a way that the Riemannian dis-

tance between nodes is almost the same as the

distance between nodes given in terms of steps

along the graph edges.

Also the notion of locality, presented in sec-

tion 7, comes with its own diﬃculties, that can be

overcome by proving a generalisation of the result

known as “unitarity plus causality implies local-

izability" [32]. This is where the theory, which is

inspired by that of QCAs, deeply diﬀers from the

theory of classical cellular automata. In particu-

lar, considering the collection of transformations

V A V

−1

for all A acting on a given system g, we

will deﬁne the neighbourhood of g as the set of

systems on which transformations V A V

−1

act

non trivially.

Once the theory of cellular automata is fully

developed, some results are proved in Section 8.

In particular, a very useful theorem is the wrap-

ping lemma, which under very wide hypotheses—

though not universal—allows for the classiﬁcation

of automata on a given inﬁnite graph by classi-

fying automata on any suitably “wrapped" ﬁnite

version of the same graph.

In Section 9, a few examples are reviewed. In

particular, using the general notion of locality,

we apply the deﬁnition of causal inﬂuence and

the neighbourhood scheme to the case of classical

cellular automata. With the above deﬁnitions at

hand, we show that allegedly local automata are

actually non-local. This solves a long standing

puzzle about the connection between classical and

quantum automata—the so-called quantisation of

classical automata [31, 48].

The paper is concluded by Section 10 with a

summary of the results and some closing remarks.

3 Operational Probabilistic Theories

In this section we will provide a thorough intro-

duction to OPTs, which is partly a review of the

literature, and partly presentation of new results

that will be used in the remainder. We provide

here a sketchy introduction before going through

formal deﬁnitions, and use Quantum Theory as

an illustrative example for the main notions in

the framework.

Quantum theory is about system types A

1

—namely complex Hilbert spaces H

A

that are

classiﬁed by their dimension d

A

= dim(H

A

)—and

transformations that can occur on systems as a

consequence of undergoing a test. Tests are rep-

resented by quantum instruments E

X

= {E

i

}

i∈X

,

i.e. a collection of Completely Positive (CP) maps

E

i

that sum to a trace-preserving one (a channel):

E =

P

i∈X

E

i

. The quantum operation E

i

—a CP

trace non-increasing map—represents the change

in the system occurring upon the event of a spe-

ciﬁc outcome i ∈ X in the test. In the diagram-

matic language of OPTs systems are represented

as labelled wires, and instruments or quantum op-

erations as boxes with an input and output wire,

e.g.

A

E

X

B

,

A

E

i

B

,

respecitvely. States can be considered as special

transformations where the input system is I, hav-

ing H

I

= C, i.e. d

I

= 1. Notice that the a prepa-

ration test, i.e. the most general test from I to A,

represents a probabilistic preparation procedure

where some state in the collection P

X

= {ρ

i

}

i∈X

is prepared, with the sub-normalised density ma-

trix ρ

i

occurring upon reading the outcome i ∈ X.

The probability of occurrence of ρ

i

is Tr[ρ

i

], and

ρ =

P

i∈X

ρ

i

has unit trace. Preparation tests

or states for system A are represented by special

diagrams

P

X

A

,

ρ

i

A

,

respectively. The space H

A

of Hermitian opera-

tors on H

A

, whose dimension is D

A

= d

2

A

, is the

real span of states of system A.

A POVM Q

Y

= {Q

j

}

j∈Y

is a collection of ef-

fects 0 ≤ Q

j

≤ I

A

that sum to the identity opera-

tor:

P

j∈Y

Q

j

= I

A

. This kind of test can be seen

as a quantum instrument from A to I, namely a

collection of linear functionals on the real space

spanned by density matrices, where the function-

als a

j

(ρ) are deﬁned as a

j

(ρ)

:

= Tr[ρQ

j

]. The

1

We remark that for the purpose of information pro-

cessing the type of a system is captured by the dimension

of the corresponding Hilbert space, e.g. the electron spin

is of the same type as the photon polarisation. This is

slightly diﬀerent than the usual notion of system type in

physics, where the two types in the above example are

diﬀerent.

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diagrammatic representation of POVMs and ef-

fects is the following

A

Q

Y

,

A

Q

j

,

respectively.

Composite systems, such as AB, correspond to

to the Hilbert space H

A

⊗H

B

of dimension d

AB

=

d

A

d

B

. Tests {E

i

}

i∈X

from A to B and {F

j

}

j∈Y

from B to C can be run in sequence, obtaining a

new test: {D

ij

}

(i,j)∈X×Y

where D

ij

:

= F

j

E

i

. On

the other hand, tests {E

i

}

i∈X

from A to B and

{E

j

}

j∈Y

from B to B

0

on subsystems of a compos-

ite system AA

0

can be run in parallel, obtaining

the test {D

ij

}

(i,j)∈X×Y

with D

ij

:

= E

i

⊗ F

j

. In

diagrams, we draw for quantum operations

A

F

j

E

i

C

=

A

E

i

B

F

j

C

,

AA

0

E

i

⊗ F

j

BB

0

=

A

E

i

B

A

0

F

j

B

0

,

and analogously for instruments.

Very relevant structures in the theory are the

real space H

A

of Hermitian operators on H

A

,

spanned by quantum states, the cone of positive

operators in P

A

⊆ H

A

, the convex set of states

[[A]], obtained by intersecting the positive cone

with the half-space Tr[X] ≤ 1, and the convex

set of deterministic states [[A]]

1

, obtained by in-

tersecting the positive cone with the aﬃne hy-

perplane Tr[X] = 1. Similar structures can be

generalised to the space spanned by quantum op-

erations. In the general framework of OPTs, we

will systematically refer to the generalisation of

the above concepts.

3.1 Formal framework

The framework of OPTs is meant to capture the

main traits of Quantum Theory (shared e.g. by

Classical Theory, or Fermionic Theory, etc.) sum-

marised above, and use them as deﬁning proper-

ties of a family of abstract theories that might

be candidates for an alternative representation of

elementary physical systems and their transfor-

mations. In the remainder of this section we pro-

vide a brief review of the framework of OPTs (for

reference see e.g. [15, 8, 9]). Some of the most

relevant diﬀerences with respect to the quantum

case stem from the fact that systems might not

compose with the tensor product rule.

We warn the reader that this presentation has

some elements that are slightly diﬀerent from

other reviews in the literature, and is tailored

to ease the presentation of subsequent material.

Most of the results in the present sections are

new, and their proof will be given. Those the-

orems that are not original are not proved, and

due reference is provided.

An operational theory Θ consists in i) a col-

lection Test(Θ) of tests T

A→B

X

, each labelled by

input and output letters from a collection Sys(Θ)

denoting system types, e.g. A → B—that will

be systematically omitted—and by a ﬁnite set of

outcomes X; for every pair of types A, B ∈ Sys(Θ)

the set of tests of type A → B is denoted hhA →

Bii; ii) a very basic associative rule for sequential

composition: the test T

X

∈ hhA → Bii can be fol-

lowed by the test S

Y

∈ hhA

0

→ B

0

ii if A

0

≡ B, thus

obtaining the sequential composition ST

X×Y

∈

hhA → B

0

ii; iii) a rule ⊗ : (A, B) 7→ AB for com-

posing labels in parallel, and a corresponding rule

for tests ⊗ : (S

X

, T

Y

) 7→ (S ⊗ T)

X×Y

, with the fol-

lowing properties

1. Associativity: (AB)C = A(BC).

2. For every S

X

∈ hhA → Bii and T

Y

∈ hhC →

Dii, one has S

X

⊗ T

Y

∈ hhAC → BDii. Asso-

ciativity of ⊗ holds:

(S

X

⊗ T

Y

) ⊗ W

Z

= S

X

⊗ (T

Y

⊗ W

Z

).

3. Unit: there is a label I such that IA = AI =

A for every A ∈ Sys(Θ).

4. Identity: for every A ∈ Sys(Θ), a test I

A

∈

hhA → Aii such that I

B

S

X

= S

X

I

A

= S

X

, for

every S

X

∈ hhA → Bii.

5. For every A

X

∈ hhA → Bii, B

Y

∈ hhB → Cii,

D

Z

∈ hhD → Eii, E

W

∈ hhE → Fii, one has

(B

Y

⊗ E

W

)(A

X

⊗ D

Z

) = (B

Y

A

X

) ⊗ (E

W

D

Z

).

(1)

6. Braiding: for every pair of system types A, B,

there exist tests S

AB

, S

∗

AB

∈ hhAB → BAii

such that S

∗

BA

S

AB

= S

BA

S

∗

AB

= I

AB

, and

S

AB

(A

X

⊗ B

Y

) = (B

Y

⊗ A

X

)S

AB

. Moreover,

S

(AB)C

= (I

A

⊗ S

BC

)(S

AC

⊗ I

B

),

S

A(BC)

= (S

AB

⊗ I

C

)(I

B

⊗ S

AC

).

When S

∗

AB

≡ S

AB

, the theory is symmetric.

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All the theories developed so far are symmetric.

All tests of an operational theory are (ﬁnite)

collections of events: hhA → Bii 3 R

X

= {R

i

}

i∈X

.

If hhA → Bii 3 R

X

= {R

i

}

i∈X

and hhB → Cii 3

T

Y

= {T

j

}

j∈Y

, then

hhA → Cii 3 (TR)

X×Y

:

= {T

j

R

i

}

(i,j)∈X×Y

.

Similarly, for R

X

∈ hhA → Bii and T

Y

∈ hhC →

Dii,

(R ⊗ T)

X×Y

:

= {R

i

⊗ T

j

}

(i,j)∈X×Y

.

The set of events of tests in hhA → Bii is de-

noted by [[A → B]]. By the properties of se-

quential and parallel composition of tests, one

can easily derive associativity of sequential and

parallel composition of events, as well as the ana-

logue of Eq. (1). For every test T

X

∈ hhA → Bii

with T

X

= {T

i

}

i∈X

, and every disjoint partition

{X

j

}

j∈Y

of X =

S

j∈Y

X

j

, one has a coarse grain-

ing operation that maps T

X

to T

0

Y

∈ hhA → Bii,

with T

0

Y

= {T

0

j

}

j∈Y

. We deﬁne T

X

j

:

= T

0

j

. The

parallel and sequential compositions distribute

over coarse graining:

T

X

j

⊗ R

k

= (T ⊗ R)

X

j

×{k}

,

A

l

T

X

j

B

k

= (A T B)

{l}×X

j

×{k}

.

Notice that for every test T

X

∈ hhA → Bii there

exists the singleton test T

0

∗

:

= {T

X

}. One can

easily prove that the identity test I

A

is a sin-

gleton: I

A

= {I

A

}, and I

B

T = T I

A

for

every event T ∈ [[A → B]]. Similarly, for

S

AB

= {S

AB

} and S

∗

AB

= {S

∗

AB

} we have

S

∗

BA

S

AB

= S

BA

S

∗

AB

= I

AB

. The collection of

events of an operational theory Θ will be denoted

by Ev(Θ). The above requirements make the col-

lections Test(Θ) and Ev(Θ) the families of mor-

phisms of two braided monoidal categories with

the same objects—system types Sys(Θ).

An operational theory is an OPT if the tests

hhI → Iii are probability distributions: 1 ≥ T

i

=

p

i

≥ 0, so that

P

i∈X

p

i

= 1, and given two tests

S

X

, T

Y

∈ hhI → Iii with S

i

= p

i

and T

i

= q

i

, the

following identities hold

S

i

⊗ T

j

= S

i

T

j

:

= p

i

q

j

,

T

X

j

:

=

X

i∈X

j

p

i

,

meaning that events in the same test are mutually

exclusive and events in diﬀerent tests of system I

are independent. While it is immediate that 1 ∈

[[I → I]]—since {1}

∗

= {I

I

} is the only singleton

test—we will assume that 0 ∈ [[I → I]]. This

means that we can consider e.g. tests of the form

{1, 0, 0}.

Events in [[A → B]] are called transformations.

As a consequence of the above deﬁnitions, every

set [[A]]

:

= [[I → A]] can be viewed as a set of

functionals on [[

¯

A]]. As such, it can be viewed as

a spanning subset of the real vector space [[A]]

R

of linear funcitonals on [[

¯

A]]. On the other hand

[[

¯

A]]

:

= [[A → I]] is a separating set of positive

linear functionals on [[A]], which then spans the

dual space [[A]]

∗

R

=: [[

¯

A]]

R

. The dimension D

A

of

[[A]]

R

(which is the same as that of [[

¯

A]]

R

) is called

size of system A. One can easily prove that in

any OPT Θ, I is the unique system with unit size

D

I

= 1. Using the properties of parallel compo-

sition, one can also prove that D

AB

≥ D

A

D

B

.

Events in [[A]] are called states, and denoted by

lower-case greek letters, e.g. ρ, while events in [[

¯

A]]

are called eﬀects, and denoted by lower-case latin

letters, e.g. a. When it is appropriate, we will

use the symbol |ρ) to denote a state, and (a| to

denote an eﬀect. We will also use the circuit no-

tation, where we denote states, transformations

and eﬀects by the symbols

ρ

A

,

A

B

,

A

a

,

respectively. Sequential composition of A ∈

[[A → B]] and B ∈ [[B → C]] is denoted by the

diagram

A

BA

C

=

A

B

C

.

For composite systems we use diagrams with mul-

tiple wires, e.g.

A

B

C D

.

The identity will be omitted:

A

I

A

=

A

. The swap S

AB

and its inverse will be de-

noted as follows

A

S

B

B A

=

B

A

S

∗

B

B A

=

B

A

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In the present paper we will always assume

that the theory under consideration is symmet-

ric, however all the results will be straightfor-

wardly generalisable. We will consequently draw

the swap S

AB

as

A

S

B

B A

=

A

S

∗

B

B A

=

B

A

An OPT Θ is speciﬁed by the collections of

systems and tests, along with the parallel com-

position rule ⊗

Θ ≡ (Test(Θ), Sys(Θ), ⊗).

Deﬁnition 1 (Equal transformations). Let

A , B ∈ [[A → B]]. Then we deﬁne A = B

if for every system C and every ρ ∈ [[AC]] and

a ∈ [[

¯

B

¯

C]], one has

ρ

A

B

a

C

=

ρ

A

B

a

C

Notice that, since states separate eﬀects and

viceversa eﬀects separate states, the above deﬁ-

nition is equivalent to the two following equality

criteria.

Lemma 1. Let A , B ∈ [[A → B]]. Then the

following conditions are equivalent.

1. A = B.

2. For every C and every ρ ∈ [[AC]], one has

ρ

A

B

C

=

ρ

A

B

C

.

(2)

3. For every C and every a ∈ [[

¯

B

¯

C]], one has

A

B

a

C

=

A

B

a

C

.

(3)

One can show [8] that events T ∈ [[A → B]]

can be identiﬁed with a family of linear maps,

one for every C, that characterize the action of

T ⊗ I

C

on [[AC]]

R

as a linear map to [[BC]]

R

.

As anticipated in the introductory paragraph, we

remind the reader that some of the diﬃculties

that we will be faced with in the remainder orig-

inate from the fact that in a general theory it is

not true that [[AB]]

R

= [[A]]

R

⊗ [[B]]

R

, but only

[[A]]

R

⊗[[B]]

R

⊆ [[AB]]

R

. As a consequence, the lin-

ear map representing T on [[A]]

R

is not suﬃcient

to determine the linear map representing T ⊗I

C

on [[AC]]

R

.

One can easily prove that [[A → B]] spans a real

vector space [[A → B]]

R

. Being [[A → B]] spanning

for [[A → B]]

R

, the criteria of deﬁnition 1 and

lemma 1 hold for A , B ∈ [[A → B]]

R

. Elements

of [[A → B]]

R

are called generalized events. Every

space [[A → B]]

R

has a zero event 0

A→B

:

= 0

I→I

⊗

T ∈ [[A → B]], where T is an arbitrary event in

[[A → B]]. As a consequence of the coarse graining

rule for tests on [[I → I]], one can easily show that

coarse graining of two or more transformations is

represented by their sum. Precisely, given a test

{T

i

}

i∈X

⊆ [[A → B]], for X

0

= {0, 1} ⊆ X one has

T

0

= T

0

+ T

1

, namely for every system C it is

T

0

⊗ I

C

= T

0

⊗ I

C

+ T

1

⊗ I

C

. Finally, every

set [[A → B]] has a subset consisting in singleton

events, that we denote by [[A → B]]

1

, and call

deterministic. For A, B 6= I, a deterministic event

is called channel. A channel U ∈ [[A → B]]

1

is

reversible if there exists a channel V ∈ [[B → A]]

1

such that V U = I

A

, U V = I

B

.

Let us now deﬁne the cones

[[A → B]]

+

:

= {λT | λ ≥ 0, T ∈ [[A → B]]}.

We will often write A  0 as a shorthand for

A ∈ [[A → B]]

+

. The cone [[A → B]]

+

introduces

a partial ordering in [[A → B]]

R

, deﬁned by

A  B ⇔ (A − B)  0.

OPTs are assumed to have all sets [[A → B]] (and

thus also cones [[A → B]]

+

) closed in the opera-

tional norm.

Two systems may be operationally equivalent

if they can be mapped one onto the other via a

reversible transformation. Clearly, in this case

every processing of the ﬁrst system is perfectly

simulated by a processing of the other, and vicev-

ersa.We deﬁne operationally equivalent systems

as follows.

Deﬁnition 2. Let A, B be two systems. We say

that A and B are operationally equivalent, in for-

mula A

∼

=

B, if there exists a reversible transfor-

mation U ∈ [[A → B]]

1

.

Deﬁnition 3. If A

1

and A

2

are operationally

equivalent through U and B

1

and B

2

through V ,

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then, for every system C, A

1

∈ [[A

1

C → B

1

C]]

R

and A

2

∈ [[A

2

C → B

2

C]]

R

are operationally

equivalent if A

2

= (V ⊗ I

C

)A

1

(U

−1

⊗ I

C

). In

particular, ρ

1

∈ [[A

1

C]]

R

and ρ

2

∈ [[A

2

C]]

R

are

operationally equivalent if ρ

2

= (U ⊗I

C

)ρ

1

, and

a

1

∈ [[

¯

A

1

¯

C]]

R

and a

2

∈ [[

¯

A

2

¯

C]]

R

are operationally

equivalent if a

2

= a

1

(U

−1

⊗ I

C

).

Lemma 2. Let AB be operationally equivalent to

A. Then B must be the trivial system I.

Proof. Since D

AB

≥ D

A

D

B

≥ D

A

, if AB

∼

=

A

the chain of inequalities is saturated, and thus

D

B

= 1.

3.2 Causality and the no-restriction hypothesis

In the remainder of the paper we will focus on

causal theories. The causality property, that

we deﬁne right away, characterizes those theories

where signals can propagate only in the direc-

tion deﬁned by input and output of processes,

within a cone of causal inﬂuence determined by

interactions between systems. These are the only

theories where one can consistently use informa-

tion acquired in a set of tests to condition the

choice of subsequent tests, where the partial or-

dering we are referring to is that determined by

the input/output direction.

Deﬁnition 4 (Causal theories). A theory (T , A)

is causal if for every test {T

i

}

i∈X

and every col-

lection of tests {S

i

j

}

j∈Y

labelled by i ∈ Y, the

generalized test {C

i,j

}

(i,j)∈X×Y

with

A

C

i,j

C

:

=

A

T

i

B

S

i

j

C

,

is a test of the theory.

Notice that this notion of causality is strictly

stronger than the one usually adopted in the lit-

erature about OPTs, that can be summarised

as uniqueness of the deterministic eﬀect (see

e.g. Refs. [8, 9, 15]). Indeed, a ﬁrst result of

crucial importance derives uniqueness of the de-

terministic eﬀect for every system type in causal

theories.

Theorem 1 ([8, 15]). In a causal theory, for ev-

ery system type A the set of deterministic eﬀects

[[

¯

A]]

1

is the singleton {e

A

}.

A proof of the above theorem, that proceeds

by contradiction, can be found in the mentioned

references. We remark that, being the notion of

causality in these references diﬀerent form the one

deﬁned here, the theorem has a slightly diﬀerent

statement. In the same references, one can ﬁnd

the equivalence of uniqueness of the deterministic

eﬀect with non-signalling from output to input.

Theorem 2 ([8, 15]). In an OPT, every system

type A has a unique deterministic eﬀect if and

only if the marginal probabilities of preparation

tests cannot depend on the choice of subsequent

observation tests.

As a consequence of our notion of causality, in

a non-deterministic theory (i.e. a theory where

[[I]] 6= {0, 1}), every convex combination of tests

is a test (see e.g. [15]). This implies that the sets

[[A]]

∗

, [[

¯

A]]

∗

, [[A → B]]

∗

are convex, where ∗ can be

replaced by nothing, 1 or +. Moreover, the sets

with ∗ = + are convex cones.

Another important consequence of causality is

that a transformation C ∈ [[A → B]] is determin-

istic if and only if it maps the deterministic eﬀect

e

B

to the deterministic eﬀect e

A

.

Theorem 3 ([8, 15]). In a causal theory, a trans-

formation C ∈ [[A → B]] is a channel iﬀ

A

C

B

e

=

A

e

.

In causal theories, one can always assume that

every state is proportional to a deterministic one.

Indeed, including in the theory every state ρ ∈

[[A]]

+

such that (e

A

|ρ) = 1 does not introduce any

inconsistency with the set of transformations, as

we now prove. Let Θ be a causal OPT, and deﬁne

the theory Θ

0

through the bijection κ : Sys(Θ) →

Sys(Θ

0

) :: A 7→ A

0

, with

[[A

0

→ B

0

]]

R

≡ [[A → B]]

R

,

[[A

0

→ B

0

]]

1

:

= {T  0 | (e

B

|T = (e

A

|},

[[A

0

→ B

0

]]

:

= {T  0 | ∃C ∈ [[A

0

→ B

0

]]

1

, C  T },

hhA

0

→ B

0

ii

:

= {T ⊆ [[A

0

→ B

0

]] |

X

T ∈T

T ∈ [[A

0

→ B

0

]]

1

}.

where  denotes the ordering induced by the cones

of the theory Θ, and the cardinality of sets T

in the last deﬁnition is implicitly assumed to be

ﬁnite.

Theorem 4. Let Θ be a causal OPT, and con-

sider the theory Θ

0

deﬁned above. Then Θ

0

is an

OPT with a unique deterministic eﬀect e

A

0

for

every A

0

.

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Proof. All the compositional structures of Θ are

inherited by Θ

0

. The deﬁning conditions in the

special case of [[A

0

]] = [[I

0

→ A

0

]] give

[[A

0

]] = {ρ ∈ [[A]]

+

| (e

A

|ρ) ≤ 1},

while for [[

¯

A

0

]] = [[A

0

→ I

0

]] they give

[[

¯

A

0

]]

1

= {e

A

}.

What remains to be proved is that the set of

transformations is closed under sequential and

parallel composition. First of all, we observe that

by theorem 3 [[A → B]]

1

⊆ [[A

0

→ B

0

]]

1

, thus

[[A → B]] ⊆ [[A

0

→ B

0

]]. This makes [[A → B]]

+

⊆

[[A

0

→ B

0

]]

+

. On the other hand, since by def-

inition it is also [[A

0

→ B

0

]] ⊆ [[A → B]]

+

, we

have [[A

0

→ B

0

]]

+

⊆ [[A → B]]

+

, which makes

[[A

0

→ B

0

]]

+

≡ [[A → B]]

+

. Thus the ordering  is

the same for the two theories. As a consequence,

the preservation of cones under the two composi-

tions is inherited by Θ

0

from the same property in

Θ. Now, if A ∈ [[A

0

→ B

0

]]

1

and B ∈ [[B

0

→ C

0

]]

1

,

then

(e

C

|BA = (e

B

|A = (e

A

|,

thus BA ∈ [[A

0

→ C

0

]]

1

. Moreover, thanks to

causality of the theory Θ one has e

AB

= e

A

⊗ e

B

,

thus for A ∈ [[A

0

→ B

0

]]

1

and B ∈ [[C

0

→ D

0

]]

1

,

(e

BD

|A ⊗ B = (e

B

|A ⊗ (e

D

|B

= (e

A

| ⊗ (e

C

| = (e

AC

|,

and consequently A ⊗ B ∈ [[A

0

C

0

→ B

0

D

0

]]

1

.

Now, given events A , B in the theory Θ

0

, by deﬁ-

nition we have C , D deterministic in Θ

0

such that

C  A and D  B, thus

F

:

=(D − B)(C − A ) + (D − B)A

+ B(C − A )  0,

G

:

=(C − A ) ⊗ (D − B)

+ (C − A ) ⊗ B + A ⊗ (D − B)  0.

Finally, F + BA  BA and G + A ⊗ B  A ⊗

B are channels, and thus BA and A ⊗ B are

events.

One can now show that the new theory Θ

0

is

causal.

Corollary 1. Let Θ be a causal OPT, and Θ

0

as

in theorem 4. Then Θ

0

is a causal OPT.

Proof. Let {T

i

}

i∈X

be a test in [[A

0

→ B

0

]], and

{S

i

j

}

j∈Y

be tests in [[B

0

→ C

0

]] for every i ∈ X.

Then

∀i ∈ X

X

j∈Y

(e

C

0

|S

i

j

= (e

B

0

|,

X

i∈X

(e

B

0

|T

i

= (e

A

0

|.

This implies that

X

i∈X

X

j∈Y

(e

C

0

|W

i,j

= (e

A

0

|,

W

i,j

:

= S

i

j

T

i

∈ [[A

0

→ C

0

]] ∀i ∈ X, j ∈ Y.

As a consequence, every conditional test W

i,j

:

=

S

i

j

T

i

∈ [[A

0

→ C

0

]] is admitted in the theory

Θ

0

.

Remark 1. In the remainder, we will focus on

theories satisfying causality and the further re-

quirements

[[A → B]]

1

= {T  0 | (e

B

|T = (e

A

|},

[[A → B]] = {T  0 | ∃C ∈ [[A → B]]

1

, C  T },

hhA → Bii = {T ⊆ [[A → B]] |

X

T ∈T

T ∈ [[A → B]]

1

}.

In particular, this implies that [[A]] = {ρ ∈ [[A]]

+

|

(e

A

|ρ) ≤ 1}. Thanks to corollary 1, this is not a

signiﬁcant restriction.

We now narrow down focus on theories that

satisfy the no-restriction hypothesis. Let us con-

sider the set of preparation-tests for every system

of the theory Θ. Then we will complete the set

of tests allowing for all those tests that transform

preparation tests to preparation tests, even when

applied locally. This requirement is the general-

isation of the assumption made in quantum the-

ory that a map is a transformation if and only

if it is completely positive and trace non increas-

ing, and a collection of transformations is a test

if and only if the sum of its elements is a channel,

i.e. completely positive and trace-preserving.

Assumption 1. Let A ∈ [[A → B]]

R

. The no-

restriction hypothesis consists in the requirement

that T ∈ hhA → Bii if and only if for every C and

every P ∈ hhI → ACii, one has (T ⊗ I

C

)P ∈ hhI →

BCii.

We will write A  0 if for every C and every

P ∈ [[AC]]

+

, one has (A ⊗ I

C

)P ∈ [[BC]]

+

. The

set of transformations A  0 is a cone, that we

denote by

K(A → B)

:

= {A ∈ [[A → B]]

R

| A  0}.

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The above deﬁned cone introduces a (partial) or-

dering in [[A → B]]

R

, deﬁned by

A  B ⇔ A − B  0. (4)

Notice that, under the no-restriction hypothe-

sis, the following identity holds

K(A → B) = [[A → B]]

+

,

and thus for every A , B ∈ [[A → B]]

R

,

A  B ⇔ A  B. (5)

We remark that, under assumption 1, the hy-

pothesis of theorem 3 can be relaxed to C ∈

[[A → B]]

+

, since the latter implies that C = λC

0

for some C

0

∈ [[A → B]] and λ ≥ 0. Then,

C  0. Moreover, since C sends e

B

to e

A

,

one has (C ⊗ I

C

)[[AC]]

1

⊆ [[BC]]

1

, and thus

(C ⊗ I

C

)[[AC]] ⊆ [[BC]]. By the no-restriction

hypothesis, this is tantamount to C ∈ [[A → B]].

We can now prove a theorem that is a very

important consequence of the no-restriction hy-

pothesis, along with causality.

Theorem 5. In a theory satisfying the no-

restriction hypothesis, let T ∈ [[A → B]]

R

. Then

T , C − T  0 for some channel C ∈ [[A → B]]

1

iﬀ T ∈ [[A → B]].

Proof. The hypothesis T , C − T  0 is equiva-

lent to T , C − T ∈ [[A → B]]

+

. Equivalently,

for every C and every P ∈ [[AC]]

+

, one has

(X ⊗ I

C

)P ∈ [[BC]]

+

for X = T , C − T . Let

now P ∈ [[AC]]. Then one has

(e

BC

|[(C − T ) ⊗ I

C

]|P ) ≥ 0,

and thus

0 ≤ (e

BC

|(T ⊗ I

C

)|P ) ≤ (e

AC

|P ) ≤ 1.

This implies that (T ⊗ I

C

)P ∈ [[AC]]. Finally,

by the no-restriction hypothesis, T ∈ [[A → B]].

The converse statement is trivial.

In the literature [10, 11] one can often ﬁnd a dif-

ferent requirement under the name “no-restriction

hypothesis”, namely that for every system A the

set [[

¯

A]] coincides with the set of functionals a

on [[A]]

R

that satisfy 0 ≤ (a|ρ) ≤ 1 for ev-

ery ρ ∈ [[A]]. The last condition may be nei-

ther necessary nor suﬃcient for the no-restriction

hypothesis as we state it, despite counterexam-

ples are still unknown for both cases. Indeed,

the no-restriction hypothesis for eﬀects imposes

the requirement that for every C and every state

P ∈ [[AC]], one has

P

A

a

C

∈ [[C]] . (6)

3.3 Norms

As the ﬁrst step in the present work is to con-

struct inﬁnite composite systems, we will need a

thorough notion of sequences and limits. From a

topological point of view the vector spaces that

we constructed so far have no special structure,

however the operational procedures for discrim-

ination of processes provide a natural deﬁnition

of distance between events. Such a distance is re-

lated to the success probability in discriminating

events. Making the space of events into a metric

space, the operational distance immediately pro-

vides a topological structure through the induced

operational norm. However, in order to make the

space of events of inﬁnite composite systems into

a Banach algebra, a stronger norm is needed, that

is introduced here: the sup-norm. In the quan-

tum and classical case the two norms coincide.

The operational norm k·k

op

for states is deﬁned

as follows [15].

Deﬁnition 5. The operational norm on [[A]]

R

is

kρk

op

:

= sup

a∈[[

¯

A]]

(2a − e|ρ) = sup

a

0

,a

1

∈[[

¯

A]]

a

0

+a

1

=e

A

(a

0

− a

1

|ρ).

(7)

The operational norm k·k

op

for transformations

A ∈ [[A → B]]

R

on ﬁnite systems is then deﬁned

as

kA k

op

:

= sup

C,Ψ∈[[AC]]

1

k(A ⊗ I

C

)Ψk

op

= sup

C,Ψ∈[[AC]]

sup

a∈[[

¯

B

¯

C]]

(2a − e|

˜

A ⊗ I

C

|Ψ).

In the special case of eﬀects a ∈ [[

¯

A]]

R

we have

kak

op

:

= sup

C,Ψ∈[[AC]]

k(a ⊗ I

C

)Ψk

op

= sup

C,Ψ∈[[AC]]

sup

b∈[[

¯

C]]

(a ⊗ [2b − e

C

]|Ψ).

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As a consequence,

kak

op

= sup

ρ

0

,ρ

1

∈[[A]]

ρ

0

+ρ

1

∈[[A]]

(a|ρ

0

− ρ

1

)

= sup

p∈[0,1]

{p sup

ρ∈[[A]]

(a|ρ) − (1 − p) inf

σ∈[[A]]

(a|σ)}

= max{ sup

ρ∈[[A]]

(a|ρ), inf

σ∈[[A]]

(a|σ)}

= sup

ρ∈[[A]]

|(a|ρ)|.

Here we only prove one new result about the op-

erational norm, that we will use in the following.

Lemma 3. Let ρ ∈ [[A]]

R

and σ ∈ [[B]]

1

. Then

kρ ⊗ σk

op

= kρk

op

. (8)

Proof. By deﬁnition it is

kρ ⊗ σk

op

= sup

(a

0

,a

1

)

(a

0

− a

1

|ρ ⊗ σ)

= sup

(b

0

,b

1

)∈M

(b

0

− b

1

|ρ),

where M

:

= {(b

0

, b

1

) | b

i

= a

i

(I

A

⊗ σ)}. Thus

kρ ⊗ σk

op

≤ kρk

op

.

On the other hand, for every binary observation-

test (a

0

, a

1

) in [[

¯

A]] one has

a

i

= (a

i

⊗ e

B

)(I

A

⊗ σ),

and thus M actually contains every possible bi-

nary observation-test on A. Finally, this implies

that

kρ ⊗ σk

op

= kρk

op

.

For more details on k·k

op

see Refs. [8, 15].

We now proceed to deﬁne the sup-norm.

Deﬁnition 6. The sup-norm kA k

sup

of A ∈

[[A → B]]

R

is deﬁned as

kA k

sup

:

= inf J(A ),

J(B)

:

= {λ | ∃C ∈ [[A → B]]

1

, λC  B  −λC }.

We now show that k·k

sup

actually deﬁnes a

norm.

Proposition 1. The sup-norm on [[A → B]]

R

is

well deﬁned:

1. kA k

sup

is non-negative, and it is null iﬀ

A = 0,

2. for µ ∈ R one has kµA k

sup

= |µ|kA k

sup

,

3. kA + Bk

sup

≤ kA k

sup

+ kBk

sup

.

Proof. 1. Let A ∈ [[A → B]]

R

. Suppose that

j

:

= inf J(A ) < 0. Then there exists 0 < ε < |j|

such that j + ε ∈ J(A ), namely

− (|j| − ε)C  A  (|j| − ε)C

⇒ −2(|j| − ε)C  0,

for some C ∈ [[A → B]]

1

, which is absurd. Then

inf J(A ) ≥ 0. Suppose now that J(A ) = 0.

This implies that for every n ∈ N there exists

C

n

∈ [[A → B]]

1

such that

1

n

C

n

 A  −

1

n

C

n

.

Now, by the closure of [[A]]

+

in the operational

norm, and considering that the sequences (

1

n

C

n

±

A ) converge to ±A , it must be ±A ∈ [[A]]

+

.

This is possible if and only if A = 0. 2. The

proof is trivial for µ = 0. Let then µ 6= 0. If

x ∈ J(A ), then xC ± A  0 for some C ∈

[[A → B]]

1

, and thus |µ|(xC ± A )  0, namely

|µ|x ∈ J(µA ). Thus kµA k

sup

≤ |µ|kA k

sup

. For

the same reason, kA k

sup

≤ (1/|µ|)kµA k

sup

, and

ﬁnally kµA k

sup

= |µ|kA k

sup

. 3. Let now x ∈

J(A ) and y ∈ J(B). Then xC ± A  0, and

yD ± B  0, for C , D ∈ [[A → B]]

1

. Thus (x +

y)F ± (A + B)  0, where F

:

= x/(x + y)C +

y/(x+y)D ∈ [[A → B]]

1

. Thus, x+y ∈ J(A +B),

and then kA + Bk

sup

≤ kA k

sup

+ kBk

sup

.

The following property makes ([[A →

A]]

R

, k·k

sup

) a Banach algebra.

Proposition 2. For A ∈ [[B → C]]

R

and B ∈

[[A → B]]

R

, kA Bk

sup

≤ kA k

sup

kBk

sup

.

Proof. Let x ∈ J(A ), and y ∈ J(B). Then there

are C , D such that xC ± A  0, yD ± B  0.

Now, we have

1

2

[(xC + A )(yD − B) + (xC − A )(yD + B)

= xyC D − A B  0

1

2

[(xC + A )(yD + B) + (xC − A )(yD − B)

=xyC D + A B  0,

and then xy ∈ J(A B). Thus, kA Bk

sup

≤

kA k

sup

kBk

sup

.

Lemma 4. Let A ∈ [[A → B]]

R

, and for an arbi-

trary C, let D ∈ [[C → D]]

1

. Then kA ⊗ Dk

sup

=

kA k

sup

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 11

Proof. Let x ∈ J(A ). Then by deﬁnition there

exists C ∈ [[A → B]]

1

such that xC ± A  0.

This implies that

xC ⊗ D ± A ⊗ D  0,

namely x ∈ J(A ⊗ D), and then J(A ) ⊆ J(A ⊗

D). Now, let y ∈ J(A ⊗ D). Then there exists

C

0

∈ [[AC → BD]]

1

such that yC

0

± A ⊗ D  0.

Composing the l.h.s. of the latter relation with

ψ ∈ [[C]]

1

and e

D

, we obtain yC

00

± A  0,

where C

00

:

= (e

D

|C

0

|ψ) is a channel, and then

y ∈ J(A ). Thus, J(A ⊗ D) ⊆ J(A ). Finally,

since J(A ⊗ D) = J(A ), we have kA ⊗ Dk

sup

=

kA k

sup

.

Corollary 2. For A ∈ [[A → B]]

R

, and for arbi-

trary C, it is kA ⊗ I

C

k

sup

= kA k

sup

.

Corollary 3. For A ∈ [[A → B]]

R

and B ∈

[[C → D]]

R

, kA ⊗ Bk

sup

≤ kA k

sup

kBk

sup

.

Proof. The result follows straightforwardly from

proposition 2 and corollary 2.

An important result regarding the sup-norm is

provided by the following proposition.

Proposition 3. For A ∈ [[B → C]]

R

and B ∈

[[A → B]]

R

, kA Bk

op

≤ kA k

sup

kBk

op

.

Proof. By deﬁnition we have

kA Bk

op

= sup

D,Ψ∈[[AD]]

1

sup

a∈[[

¯

C

¯

D]]

(2a − e

CD

|A B ⊗ I

D

|Ψ).

Now, for every a ∈ [[

¯

C

¯

D]], and λ ∈ J(A ), upon

deﬁning a

0

:

= a, a

1

:

= e

CD

− a, we have

(a

0

− a

1

|A ⊗ I

C

= (a

0

|[A ⊗ I

D

] − (a

1

|[A ⊗ I

D

]

= λ[(˜a

0

| − (˜a

1

|],

where (˜a

i

|

:

= λ

−1

{(a

i

|[A ⊗I

D

]+

1

2

(e|[(λC −A )⊗

I

D

]}, with C : [[B → C]]

1

such that λC ± A  0.

Clearly,

˜a

0

, ˜a

1

∈ [[

¯

B

¯

D]], ˜a

0

+ ˜a

1

= e

BD

,

thus

(a

0

− a

1

|A B ⊗ I

D

|Ψ)

= λ[(˜a

0

| − (˜a

1

|]B ⊗ I

D

|Ψ)

≤ λ sup

b∈[[

¯

B

¯

D]]

(2b − e

BD

|B ⊗ I

D

|Ψ)

This implies that kA Bk

op

≤ λ sup

b∈[[

¯

B

¯

D]]

(2b −

e

BD

|B ⊗ I

D

|Ψ), and then for every λ ∈ J(A )

kA Bk

op

≤ λ sup

D,Ψ∈[[AD]]

1

sup

b∈[[

¯

B

¯

D]]

(2b − e

BD

|B ⊗ I

D

|Ψ)

= λkBk

op

.

Finally, taking the inﬁmum over λ ∈ J(A ) we

get

kA Bk

op

≤ kA k

sup

kBk

op

.

Corollary 4. The sup-norm is stronger than the

operational norm.

Proof. It is suﬃcient to observe that kA k

op

=

kA I k

op

≤ kA k

sup

kI k

op

= kA k

sup

.

Lemma 5. Let A ∈ [[A → B]]

1

be a channel.

Then kA k

sup

= 1.

Proof. Clearly, A  A  −A , thus 1 ∈ J(A ),

and inf J(A ) ≤ 1. On the other hand, suppose

that there exists 1 > λ ∈ J(A ). This implies

that there exists a channel C ∈ [[A → B]]

1

such

that D

:

= λC − A  0. However, this implies

that (e|

B

D = −(1 − λ)(e|

A

 0. This is ab-

surd, and then λ ∈ J(A ) must be λ ≥ 1. Thus,

inf J(A ) ≥ 1. Finally, this implies kA k

sup

=

1.

Corollary 5. The sup-norm of the identity chan-

nel I ∈ [[A → A]]

1

is 1.

The sup-norm for eﬀects is just the special case

of the sup-norm of transformations with the out-

put system equal to I.

Deﬁnition 7. Let a ∈ [[

¯

A]]

R

, and let us deﬁne

the half-line

J(a)

:

= {λ ∈ R

+

| λe

A

 a  −λe

A

}.

The sup-norm kak

sup

is deﬁned as

kak

sup

:

= inf J(a).

Proposition 4. The sup-norm on [[

¯

A]]

R

is well

deﬁned:

1. kak

sup

is non-negative, and it is null iﬀ a =

0,

2. for µ ∈ R one has kµak

sup

= |µ|kak

sup

,

3. ka + bk

sup

≤ kak

sup

+ kbk

sup

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 12

Proof. The case of eﬀects is just a special case of

the result of Proposition 1.

As an immediate consequence of proposition 3,

we have the following result.

Corollary 6. The sup-norm is stronger than the

operational norm on [[

¯

A]]

R

.

In the special case of [[I → I]]

R

= R, one has

[[I → I]]

1

= {1}, and [[I → I]]

+

= R

+

, thus J(x) =

{λ ∈ R | λ±x ≥ 0}. Clearly, kxk

sup

= inf J(x) =

|x|. We now need the following lemma.

Lemma 6. Let a ∈ [[

¯

A]]

R

. Then ka ⊗ e

B

k

op

=

kak

op

and ka ⊗ e

B

k

sup

= kak

sup

.

Proof. For k·k

op

, the equality trivially follows

from the deﬁnition. For k·k

sup

, it is a special

case of lemma 4.

From lemma 5, the following consequence fol-

lows.

Corollary 7. For the deterministic eﬀect e

A

one

has ke

A

k

sup

= ke

A

k

op

= 1.

We now prove a result that will be very useful

later.

Lemma 7. Let A ∈ [[A → B]]

+

, and (a|

:

=

(e

B

|A ∈ [[

¯

A]]

+

. Then kA k

sup

= kak

sup

.

Proof. First of all, by proposition 2, one has

kak

sup

≤ ke

A

k

sup

kA k

sup

= kA k

sup

. On the

other hand, let λ ∈ J(a). This implies that

b

:

= λe

A

− a  0.

Let us then deﬁne B

:

= |ρ)(b| ∈ [[A → B]]

+

, for

some ρ ∈ [[B]]

1

. By theorem 3, it is easy to check

that A + B = λC for C ∈ [[A → B]]

1

. Then,

λC − A = B  0, (9)

which means that λ ∈ J(A ). Then, J(a) ⊆

J(A ), and ﬁnally kA k

sup

≤ kak

sup

.

In the case of Classical and Quantum Theory,

phrasing the deﬁnitions of sup- and operational

norm in terms of semideﬁnite programming prob-

lems, one can conclude that they are strongly

dual. As a consequence, in these special cases

they deﬁne the same norm.

4 The quasi-local algebra in OPTs

Following the deﬁnition of Ref. [31], we de-

ﬁne a cellular automaton in a general OPT as

a triple (G, A, V ), where G is a denumerable

set of labels for the systems that compose the

automaton—addresses of the memory cells—, A

G

is a (possibly inﬁnite) composite system corre-

sponding to the collection of systems A

g

labelled

by elements g ∈ G—i.e. the memory array—, and

V is a reversible transformation on [[

¯

A

G

]], such

that V · V

−1

is an automorphism of [[A

G

→ A

G

]].

However, this is deﬁnition is incomplete, for many

reasons. In the ﬁrst place, most of the objects

mentioned above are not thoroughly deﬁned. The

purpose of the present section is to set the ground

for the rigorous deﬁnition of a cellular automaton.

We start ﬁxing some notation. For every R ⊆

G let

A

R

:

=

O

g∈R

A

g

,

with the convention that A

∅

:

= I. The full sys-

tem is then A

G

=

N

g∈G

A

g

. This purely formal

notion will now be thoroughly substantiated.

With a slight abuse of notation, we will often

use R = g instead of R = {g}, dropping the

braces. The set of ﬁnite regions of G will be de-

noted as

R

(G)

:

= {R ⊆ G | |R| < ∞},

while the set of arbitrary regions of G will be

denoted as

R

(G)

:

= {R ⊆ G}.

Clearly R

(G)

⊆ R

(G)

. In the remainder, when we

write e.g. e

R

or I

R

for R ∈ R

(G)

, we mean the

deterministic eﬀect or the identity transformation

for the system A

R

, respectively.

The following construction of

N

g∈G

A

g

is in-

spired by that of Refs. [49, 50], however with sig-

niﬁcant diﬀerences.

4.1 Quasi-local eﬀects

In this subsection we start the mathematical con-

struction of the system A

G

, the parallel compo-

sition of inﬁnitely many ﬁnite systems. The ﬁrst

object that we will deﬁne is the space of gener-

alised eﬀects, along with its convex cone of pos-

itive eﬀects, and the convex set of eﬀects. The

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 13

construction is based on the notion of a local ef-

fect, that is an eﬀect which acts non trivially only

on ﬁnitely many systems labelled by g ∈ R, with

R ∈ R

(G)

. We will say that such an eﬀect acts

on the ﬁnite region R. We will then introduce a

real vector space structure over the set of local

eﬀects, and a norm that is induced by the sup

norm for local eﬀects. The last step is then the

topological closure of the space of local eﬀects in

the sup-norm. The Banach space thus obtained

is the space of generalised quasi-local eﬀects, and

the positive cone along with the convex set of ef-

fects contain the limits of sequences of elements of

local cones or convex sets of eﬀects, respectively.

As we will see in the next section, the deﬁnition

of the space of quasi-local states is much more in-

volved than that of quasi-local eﬀects. The latter

is particularly simple, thanks to causality, that

provides us with a preferential local eﬀect, the

unique deterministic one, without the need of in-

troducing any arbitrary choice of a reference local

eﬀect. Moreover, while the construction of the

space of quasi-local states is not logically nec-

essary, as we will ﬁnd them as a subspace of

bounded functionals on quasi-local eﬀects, the

same is not true of eﬀects.

If we deﬁned eﬀects as the dual space of quasi-

local states, we would end up with far more eﬀects

than needed, while lacking sectors of the state

space, that are usually reached by the evolution of

a quasi-local state through a cellular automaton.

These are the main reasons why our construction

begins with eﬀects.

The ﬁrst notion we introduce is that of a lo-

cal eﬀect. Let A

G

denote the formal composition

of countably many systems from a general OPT:

A

G

:

=

N

g∈G

A

g

. Intuitively speaking, a local ef-

fect of A

G

is an event within a test that discards

all the systems in G but the ﬁnite region R, where

a non-trivial measurement is performed. We then

deﬁne the local eﬀect (a, R) as a pair made of an

eﬀect a ∈ [[

¯

A

R

]] and the region R. To lighten the

notation, in the remainder of the paper we will

use the symbol a

R

instead of (a, R). The set of

local eﬀects is denoted by

Pre[[

¯

A

G

]]

L

:

=

G

R∈R

(G)

[[

¯

A

R

]] = {a

R

| R ∈ R

(G)

, a ∈ [[

¯

A

R

]]}.

The above deﬁnition can be widened encompass-

ing generalised eﬀects:

Pre[[

¯

A

G

]]

LR

:

=

G

R∈R

(G)

[[

¯

A

R

]]

R

.

Let us now consider a partition of the region R

into two disjoint regions S

0

∪ S

1

= R. Since in

the deﬁnition of a local eﬀect a

R

we did not set

constraints on the eﬀect a ∈ [[

¯

A

R

]]

R

, it might be

that a = e

S

0

⊗ a

0

, with a

0

∈ [[

¯

A

S

1

]]

R

. It is clear

from our intuitive notion of a local eﬀect that a

R

and a

0

S

1

should represent the same eﬀect. This

observation leads us to the equivalence relation

deﬁned as follows.

Deﬁnition 8 (Equivalent local eﬀects). We say

that the eﬀects a

R

and a

0

S

in Pre[[

¯

A

G

]]

LR

are

equivalent, and denote this as a

R

∼ a

0

S

, if there

exists a

0

∈ [[

¯

A

R∩S

]]

R

such that the following iden-

tities hold

(

a = a

0

⊗ e

S\R

,

a

0

= a

0

⊗ e

R\S

.

(10)

It is clear that the real notion of a local ef-

fect is captured by the equivalence classes modulo

the above equivalence relation. We thus quotient

Pre[[

¯

A

G

]]

R

and deﬁne the obtained set as the set

of local eﬀects of A

G

.

Deﬁnition 9. A generalised local eﬀect is an

equivalence class [a

R

]

∼

. The set of generalised

local eﬀects of A

G

is

[[

¯

A

G

]]

LR

:

= Pre[[

¯

A

G

]]

LR

/ ∼ .

With a slight abuse of notation, in the following

we will write a

R

instead of [a

R

]

∼

, unless the con-

text requires explicit distinction of the two sym-

bols. One can easily prove the following result

Lemma 8. Let a

R

∈ Pre[[

¯

A

G

]]

LR

. Then for every

ﬁnite region H ∈ R

(G)

such that H ∩ R = ∅,

one has (a ⊗ e

H

)

R∪H

∈ Pre[[

¯

A

G

]]

LR

, and (a ⊗

e

H

)

R∪H

∼ a

R

.

The proof of the above lemma is straightfor-

ward, and we do not report it here.

Let us now come back to our initial goal, that

is to deﬁne a local eﬀect as an event that acts

non-trivially only on a ﬁnite region R. Intuition

leads again to ﬁgure out what is the preferred

representative of the class of a local eﬀect: within

the equivalence class, it is the element deﬁned on

the smallest region. We now provide a formal

deﬁnition of such a minimal representative, and

show that it is well posed.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 14

Deﬁnition 10. The minimal representative of

the equivalence class a

R

, denoted as ˜a

R

a

, is de-

ﬁned through

R

a

:

=

\

S∈R

(a,R)

S, ˜a

R

a

∼ a

R

, (11)

where R

(a,R)

is the set of all those ﬁnite regions

S ∈ R

(G)

for which there exists b ∈ [[

¯

A

S

]]

R

such

that b

S

∼ a

R

.

Lemma 9. The minimal representative exists

and is unique.

Proof. As to existence, we remark that, by

Eq. (11), R

a

is the set of all g ∈ G such that

for all regions S ∈ R

(a,R)

one has g ∈ S. Thus,

if h 6∈ R

a

, there must exist S ∈ R

(a,R)

such that

h 6∈ S. This implies that i) by deﬁnition there

exists f

S

∼ a

R

, and ii) by lemma 8 c

S∪h

∼ a

R

,

with

c = f ⊗ e

h

. (12)

As a consequence, if c

T

∼ a

R

and h ∈ T , but

h 6∈ R

a

, by deﬁnition (10) c

T

must be of the

form of Eq. (12), for some f ∈ [[

¯

A

T \h

]]

R

. Clearly,

R

a

∈ R

(G)

, since for any S ∈ R

(a,R)

one has R

a

⊆

S. Now, let S ∈ R

(a,R)

, and c

S

∼ a

R

. One has

S = R

a

∪ S

0

, where S

0

:

= (S \ R

a

) ∈ R

(G)

, thus

R

a

∩ S

0

= ∅. By Eq. (12), we then have

c = b ⊗ e

S

0

, b ∈ [[

¯

A

R

a

]]

R

. (13)

We now deﬁne ˜a

R

a

:

= b

R

a

, and ﬁnally, since

Eq. 13 holds for any c

S

∼ a

R

, one can easily

verify that ˜a

R

a

∼ a

R

.

As to uniqueness, we remark that the re-

gion R

a

is uniquely deﬁned. Now, suppose that

there were two diﬀerent b, c ∈ [[

¯

A

R

a

]]

R

such that

b

R

a

, c

R

a

∼ a

R

. Then by Eq. (10) it must be

b = c.

We now make local eﬀects into a vector space.

Deﬁnition 11. Let a

R

, b

S

∈ [[

¯

A

G

]]

LR

, and h ∈ R.

Then we deﬁne

ha

R

:

=

(

(ha)

R

h 6= 0,

0

∅

h = 0,

a

R

+ b

S

:

= c

R∪S

c

:

= a ⊗ e

S\R

+ b ⊗ e

R\S

.

Notice that it is not always true that R

c

= R

a

∪

R

b

. As an example, consider a = f

g

1

⊗f

g

2

and b =

f

g

1

⊗ (e − f

g

2

), with f

g

1

6= e

g

1

, 0 6= f

g

2

6= e

g

2

, and

R

a

= R

b

= {g

1

, g

2

}. Then c = a + b = f

g

1

⊗ e

g

2

,

and clearly R

c

= {g

1

}, which is strictly included

in R

a

= R

b

= R

a

∪ R

b

.

It is easy to check that [[

¯

A

G

]]

LR

is a real vector

space with null element given by the equivalence

class of 0

∅

.

We now equip the real vector space of local

eﬀects with a norm, and we then close it to ob-

tain the Banach space of quasi-local eﬀects. The

deﬁnition is based on a norm for eﬀects of ﬁnite

systems, as every local eﬀect a

R

∈ [[

¯

A

G

]]

LR

re-

duces to the eﬀect ˜a ∈ [[

¯

A

R

a

]]

R

. A natural norm

one might think of is then the operational norm,

that induces the following deﬁnition.

Deﬁnition 12. The operational norm ka

R

k

op

of

a

R

∈ [[

¯

A

G

]]

LR

is deﬁned by the following expres-

sion

ka

R

k

op

:

= k˜ak

op

, (14)

where ˜a ∈ [[

¯

A

R

a

]]

R

.

Unfortunately, if one completes the space of

local eﬀects in the operational norm, in general

the dual norm on the space of bounded linear

functionals—which will be our state space—does

not coincide with the operational norm on the

state space, for those states that can be inter-

preted as quasi-local preparations. We will then

choose a diﬀerent norm on our space of eﬀects.

The new norm will be referred to as sup-norm, as

it is the extension of the sup-norm to the inﬁnite

case.

As far as ﬁnite-dimensional systems are con-

cerned, this choice does not represent a problem,

as all norms are equivalent in ﬁnite-dimensional

vector spaces. However, the sup-norm is stronger

than the operational one—see corollary 4—, and

thus the space that we construct, completing our

normed vector space of local eﬀects with sup-

norm Cauchy sequences, might contain distinct

limits that are operationally equivalent. This

point is a delicate one, and to avoid an unreason-

able construction where there exist diﬀerent ef-

fects that are operationally equivalent, we impose

a suﬃcient constraint for the operational norm

and the sup-norm to be equivalent also for inﬁ-

nite systems: we restrict attention to those the-

ories where the following property holds: there

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 15

exists a ﬁnite constant k such that for every sys-

tem A and every a ∈ [[

¯

A]]

R

kak

sup

≤ kkak

op

. (15)

This implies that not only the bound (15) holds

for a ﬁxed system A, but it holds with a ﬁxed

constant independent of the system A. In turn,

a suﬃcient condition for (15) is that for every

system A

[[A]]

∗

+

≡ [[

¯

A]]

+

, (16)

namely every positive functional on the cone of

states is proportional to an eﬀect by a positive

constant, and viceversa. In all the presently

known theories the latter condition is satisﬁed.

The two above conditions in Eqs. (15) and (16)

are discussed in detail in Appendix A, where we

prove that condition (16) implies condition (15).

Our new norm is an order-unit norm. As we

will discuss in subsection 4.2, its dual coincides

with the operational norm on quasi-local states.

Let us now see the deﬁnition of the sup-norm

in detail.

Deﬁnition 13. The sup-norm ka

R

k

sup

of a

R

∈

[[

¯

A

G

]]

LR

is deﬁned by the following expression

ka

R

k

sup

:

= k˜ak

sup

, (17)

where ˜a ∈ [[

¯

A

R

a

]]

R

.

We now want to prove that the operational

norm and the sup-norm are well-deﬁned norms

on [[

¯

A

G

]]

LR

.

Proposition 5. Let a

R

∼ ˜a

R

a

. Then ka

R

k

op

=

kak

op

, and ka

R

k

sup

= kak

sup

.

Proof. Let a

R

∼ ˜a

R

a

, and R = R

a

∪ R

0

, with

R

a

∩ R

0

= ∅. Then by Eq. (10) we have

a = ˜a ⊗ e

R

0

,

and thus by deﬁnitions 12 and 13 and lemma 6,

it is

ka

R

k

op

= k˜ak

op

= k˜a ⊗ e

R

0

k

op

= kak

op

,

ka

R

k

sup

= k˜ak

sup

= k˜a ⊗ e

R

0

k

sup

= kak

sup

.

We can then prove the desired result.

Proposition 6. The functionals k·k

op

and k·k

sup

are norms on [[

¯

A

G

]]

LR

.

Proof. 1. For every a

R

∈ [[

¯

A

G

]]

LR

it is clear that

ka

R

k

op

≥ 0 and ka

R

k

sup

≥ 0. Now, k˜ak

op

=

k˜ak

sup

= 0 if and only if ˜a = 0, namely, re-

minding deﬁnition 11, a

R

= 0

∅

. 2. For every

a

R

∈ [[

¯

A

G

]]

LR

, by deﬁnitions 7, 11, and 12, one

straightforwardly has that, for every µ ∈ R,

kµa

R

k

op

= kµ˜ak

op

= |µ|k˜ak

op

= |µ|ka

R

k

op

,

kµa

R

k

sup

= kµ˜ak

sup

= |µ|k˜ak

sup

= |µ|ka

R

k

sup

.

3. Let now c

R∪S

= a ⊗ e

S\R

+ e

R\S

⊗ b for

a ∈ [[

¯

A

R

]]

R

and b ∈ [[

¯

A

S

]]

R

, as from deﬁnition 11.

Then, by lemma 6, proposition 5 and by the tri-

angle inequality, we have

ka

R

+ b

S

k

op

= kc

R∪S

k

op

= kck

op

≤ kak

op

+ kbk

op

= ka

R

k

op

+ kb

S

k

op

,

ka

R

+ b

S

k

sup

= kc

R∪S

k

sup

= kck

sup

≤ kak

sup

+ kbk

sup

= ka

R

k

sup

+ kb

S

k

sup

.

We remark that, in a theory that satisﬁes as-

sumption (16), one has k·k

op

≡ k·k

sup

on [[

¯

A]]

R

,

and thus k·k

op

= k·k

sup

on [[

¯

A

G

]]

R

(the proof can

be found in appendix A). In general, however, the

sup-norm is stronger than the operational norm,

as we now prove.

Lemma 10. Let a

R

∈ [[

¯

A

G

]]

LR

. Then ka

R

k

op

≤

ka

R

k

sup

.

Proof. Let us consider µ ∈ J(a). Then for every

ρ ∈ [[A

R

]]

1

we have

(µe

R

± a|ρ) ≥ 0,

i.e. |(a|ρ)| ≤ µ. Then, taking the supremum over

states on l.h.s. we obtain ka

R

k

op

≤ µ, and ﬁnally,

taking the inﬁmum over J(a) we obtain the the-

sis.

The space of local eﬀects that we constructed

so far is a normed real vector space. We now

make it into a Banach space, the space of quasi-

local eﬀects, by the usual completion procedure

for normed spaces. We ﬁrst introduce the space

[[

¯

A

G

]]

CR

of Cauchy sequences

a : N → [[

¯

A

G

]]

LR

:: n 7→ a

n

R

n

.

We then deﬁne the equivalence relation between

Cauchy sequences in [[

¯

A

G

]]

CR

deﬁned by a

∼

=

b

iﬀ lim

n→∞

ka

n

R

n

− b

n

S

n

k

sup

= 0. Finally, we de-

ﬁne the space [[

¯

A

G

]]

QR

of generalised quasi-local

eﬀects, by taking the quotient

[[

¯

A

G

]]

QR

:

= [[

¯

A

G

]]

CR

/

∼

=

. (18)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 16

The elements of this space will be denoted by

a = [a

n

R

n

]. In this context, the class of a con-

stant Cauchy sequence (R

n

= R and a

n

R

n

= a

R

)

will be denoted by a

R

:

= [a

n

R

n

]. Clearly, since

|ka

n

R

n

k

sup

− ka

m

R

m

k

sup

| ≤ ka

n

R

n

− a

m

R

m

k

sup

,

for a Cauchy sequence a = [a

n

R

n

] also ka

n

R

n

k

sup

is a Cauchy sequence, and we deﬁne kak

sup

:

=

lim

n→∞

ka

n

R

n

k

sup

. One can easily verify that

the sup-norm on [[

¯

A

G

]]

QR

thus deﬁned is indepen-

dent of the speciﬁc sequence within an equiva-

lence class. The space [[

¯

A

G

]]

QR

is by construction

a real Banach space, and [[

¯

A

G

]]

LR

can be identi-

ﬁed with the dense submanifold containing con-

stant sequences a

R

= [a

R

]. Clearly, in this case

kak

sup

= ka

R

k

sup

.

Deﬁnition 14. Let a ∈ [[

¯

A

G

]]

QR

. If there is a

Cauchy sequence a

n

R

n

in the class deﬁning a,

such that, for some n

0

∈ N, for every n ≥ n

0

one has ˜a

n

∈ [[

¯

A

R

n

]], then we call a a quasi-local

eﬀect. We denote the set of quasi-local eﬀects by

[[

¯

A

G

]]

Q

.

The cone [[

¯

A

G

]]

Q+

contains all elements that

are proportional to an element in [[

¯

A

G

]]

Q

by a pos-

itive constant.

The deterministic quasi-local eﬀect e

G

is the

class e

G

:

= 1

∅

.

The cone [[

¯

A

G

]]

Q+

introduces a partial ordering

in [[

¯

A

G

]]

QR

, that we denote by

a  b ⇔ a − b ∈ [[

¯

A

G

]]

Q+

.

Quasi-local eﬀects can be interpreted as eﬀects

that can be arbitrarily well approximated by local

observation procedures. The eﬀect e

G

plays an

important role, as it is the unique deterministic

eﬀect in [[

¯

A

G

]]

Q

. This will be proved shortly. Let

us start providing the ﬁrst important property of

e

G

.

Lemma 11. Let a ∈ [[

¯

A

G

]]

Q

. Then also e

G

− a ∈

[[

¯

A

G

]]

Q

.

Proof. Let a = [a

n

R

n

]. Let a

0

n

R

n

:

= (e − a

n

)

R

n

∈

[[

¯

A

R

n

]]. Then one has (a

0

n

− a

0

m

)

R

n

∪R

m

= (a

m

−

a

n

)

R

m

∪R

n

, and thus a

0

n

R

n

is a Cauchy sequence

whose class we call a

0

∈ [[

¯

A

G

]]

Q

. Finally, since

a+a

0

is the class [a

n

R

n

]+[a

0

n

R

n

] = [(a

n

+a

0

n

)

R

n

] =

[1

∅

] = e

G

, we have that a + a

0

= e

G

, namely

e

G

− a = a

0

∈ [[

¯

A

G

]]

Q

.

By the above lemma we know that not only

e

G

 a for every quasi-local eﬀect a, but also that

every quasi-local eﬀect a can be complemented

by a quasi-local eﬀect a

0

such that a + a

0

= e

G

.

In other words (a, e

G

− a) is a binary quasi-local

observation test.

We can also prove the converse result.

Lemma 12. Let a ∈ [[

¯

A

G

]]

Q+

, and e

G

− a ∈

[[

¯

A

G

]]

Q+

. Then a, e

G

− a ∈ [[

¯

A

G

]]

Q

.

Proof. Indeed, the statement is true for ﬁnite

systems thanks to the no-restriction hypothesis

(see theorem 5), and thus it holds for a given

Cauchy sequence in the class of a, namely if

a

0

n

R

n

:

= (e − a

n

)

R

n

∈ [[

¯

A

R

n

]]

+

for all n, then

a

n

, a

0

n

∈ [[

¯

A

R

n

]]. Thus, a = lim

n→∞

a

n

∈ [[

¯

A

G

]]

Q

,

and a

0

= lim

n→∞

a

0

n

= e

G

− a ∈ [[

¯

A

G

]]

Q

.

We now prove that e

G

lies in the interior of

[[

¯

A

G

]]

Q+

, namely there is a ball of radius r around

e

G

in sup-norm that is contained in [[

¯

A

G

]]

Q+

.

Lemma 13. There exists an open ball B

r

(e

G

)

of radius r in [[

¯

A

G

]]

QR

that is fully contained in

[[

¯

A

G

]]

Q+

.

Proof. Let ka − e

G

k

sup

< r < 1/2. Then, by

deﬁnition, there exist a sequence a

n

R

n

such that

lim

n→∞

a

n

= a, and n

0

∈ N such that for n ≥ n

0

ka

n

− e

G

k

sup

≤ ka

n

− ak

sup

+ ka − e

G

k

sup

< 2r.

This implies that

[2re + (a

n

− e)]

R

n

= [a

n

− (1 − 2r)e]

R

n

 0.

Now, this implies that for n ≥ n

0

it is a

n



(1 − 2r)e

R

n

 0, and thus a

n

∈ [[

¯

A

G

]]

L+

. Finally,

by deﬁnition, one obtains a ∈ [[

¯

A

G

]]

Q+

. Thus, the

ball B

r

(e

G

) is contained in [[

¯

A

G

]]

Q+

.

Corollary 8. Let a ∈ [[

¯

A

G

]]

Q+

. Then for ε > 0,

B

εr

(εe

G

+ a) ⊆ [[

¯

A

G

]]

Q+

for some r > 0.

Proof. Let b ∈ B

εr

(εe

G

+ a). Then

kb − a − εe

G

k

sup

< εr

⇔ k(b − a)/ε − e

G

k

sup

< r.

Then (b − a)/ε = c ∈ [[

¯

A

G

]]

Q+

, ad thus

b = εc + a ∈ [[

¯

A

G

]]

Q+

.

We now prove that the deterministic eﬀect e

G

allows for an extension of the deﬁning property

of the sup-norm.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 17

Lemma 14. The sup-norm of a ∈ [[

¯

A

G

]]

QR

can

be expressed as

kak

sup

= inf J(a),

J(a)

:

= {µ ≥ 0 | µe

G

 a  −µe

G

}.

Proof. In the case of local eﬀects a = a

R

the

statement is a trivial recasting of the deﬁnition

of sup-norm. Let us then consider the class a ∈

[[

¯

A

G

]]

QR

with a = [a

n

R

n

]. One has, by deﬁnition,

kak

sup

= lim

n→∞

ka

n

k

sup

. Let us consider the

sequence µ

n

:

= ka

n

k

sup

+ ε ∈ J(a

n

). Clearly,

lim

n→∞

µ

n

= kak

sup

+ ε. Moreover, since

k{(µ

n

− µ

m

)e ± (a

n

− a

m

)}

R

n

∪R

m

k

sup

≤ |ka

m

k

sup

− ka

n

k

sup

| + ka

m

− a

n

k

sup

≤ 2ε,

the sequence {(µ

n

+ ε)e ± a

n

}

R

n

converges to

(kak

sup

+ε)e

G

±a ∈ [[

¯

A

G

]]

Q+

, thus for every ε ≥ 0

one has kak

sup

+ ε ∈ J(a). This implies that

inf J(a) ≤ kak

sup

.

On the other hand, let µ ∈ J(a), then

µe

G

± a  0.

By corollary 8, we have that, for ε > 0,

(ε + µ)e

G

± a ∈ B

εr

(εe

G

+ µe

G

± a) ⊆ [[

¯

A

G

]]

Q+

.

Now, this implies that there are two open balls,

centred at (µ + ε)e

G

± a, that are fully contained

in [[

¯

A

G

]]

Q+

. There exists then n

0

∈ N such that

for n ≥ n

0

one has

(µ + ε)e

G

± a

n

 0,

namely (µ + ε) ∈ J(a

n

). Thus, J(a) ⊆ J(a

n

) − ε,

and

inf J(a) ≥ ka

n

k

sup

− ε. (19)

Finally, this implies that inf J(a) ≥ kak

sup

.

As the cone [[

¯

A

G

]]

Q+

is closed, we can easily

prove that the inﬁmum in the expression of the

sup norm is actually a minimum.

Lemma 15. Let a ∈ [[

¯

A

G

]]

QR

. Then

kak

sup

e

G

± a  0. (20)

Proof. By lemma 14, for every ε > 0 one has

(kak

sup

+ ε)e

G

± a  0.

The sequences b

±

n

:

= (kak

sup

+1/n)e

G

±a are both

Cauchy, and their limits are both in [[

¯

A

G

]]

Q+

,

then

kak

sup

e

G

± a  0.

Before concluding, we remark that the opera-

tional norm can be extended to [[

¯

A

G

]]

QR

, by sim-

ply deﬁning for a = [a

n

R

n

]

kak

op

:

= lim

n→∞

ka

n

R

n

k

op

.

Indeed, since

|ka

n

R

n

k

op

− ka

m

R

m

k

op

| ≤ ka

n

R

n

− a

m

R

m

k

op

≤ ka

n

R

n

− a

m

R

m

k

sup

,

the sequence ka

n

R

n

k

op

is Cauchy, and the limit

is well deﬁned. Moreover, also on [[

¯

A

G

]]

QR

the sup-norm is stronger than the operational

norm, as can be straightforwardly concluded from

lemma 10. However, in theories where there ex-

ists k > 0 such that k·k

sup

≤ kk·k

op

in [[

¯

A]]

R

for

every ﬁnite system A, the same inequality holds

in the limit, and the operational and sup-norm

are equivalent. In particular, this is the case un-

der assumption (16).

We now introduce a diagrammatic notation for

eﬀects in [[

¯

A]]

QR

that will make part of the subse-

quent proofs and arguments more intuitive. First

of all, we denote a quasi-local eﬀect a ∈ [[

¯

A

G

]]

QR

by the symbol

G

a

. (21)

In the case of a local eﬀect a

R

∈ [[

¯

A

G

]]

LR

, we will

draw

G

a

R

=

R

a

G\R

e

. (22)

Notice that, since for (a ⊗ e

R

1

)

R

with a ∈ [[

¯

A

R

0

]]

and R

1

:

= R \ R

0

, it is (a ⊗ e

R

1

)

R

∼ a

R

0

, we have

R

0

a

R

1

e

G\R

e

=

R

0

a

(G\R)∪R

1

e

. (23)

From the above identity, we can intuitively con-

clude that

(G\R)∪R

1

e

=

R

1

e

G\R

e

. (24)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 18

Indeed, let G = R ∪ H, with |R| < ∞. Then

[e

R

] = [1

∅

] = e

G

, which is the equation repre-

sented by the diagram in Eq. 24. Similarly, let

a

R

⊗ b

H

:

= lim

n→∞

(a ⊗ b

n

), where [b

nS

n

] = b ∈

[[

¯

A

H

]]

QR

. By corollary 3, one has [(a⊗b

n

)

R∪S

n

] =

(a

R

⊗ b

H

). Thus, in G = R ∪ H, one has

a

R

= [a

R

] = [(a

R

⊗ e

H

)] = a

R

⊗ e

H

, which is

the meaning of Eq. (22).

As a ﬁnal remark, we observe that for every

denumerable set G, and every subset R ⊆ G, in-

cluding inﬁnite ones, one can deﬁne [[

¯

A

(G)

R

]]

QR

as

the closed subspace spanned by those quasi-local

eﬀects a = a

n

R

n

such that, for every n ∈ N, R

n

⊆

R. If one constructs the system A

R

:

=

N

g∈R

A

g

,

it is straightforward to construct an ordered Ba-

nach space isomorphism

J

†

R

: [[

¯

A

(G)

R

]]

QR

→ [[

¯

A

R

]]

QR

:: [a

n

R

n

]

G

7→ [a

n

R

n

]

R

,

(25)

with the norm coinciding with the sup-norm in

both spaces. The left-inverse of J

†

R

is

J

−1†

R

: [[

¯

A

R

]]

QR

→ [[

¯

A

(G)

R

]]

QR

. (26)

J

†

R

and J

−1†

R

can be diagrammatically denoted

as

R

J

†

R

a

G\R

e

=

R

a

,

(27)

G

J

−1†

R

a

=

R

a

G\R

e

. (28)

4.2 Extended states

Now that we deﬁned the space of quasi-local ef-

fects, we can deﬁne the space of states as the

space of bounded linear functionals on [[

¯

A

G

]]

QR

.

The set of states is then deﬁned considering those

linear functionals that, acting on local eﬀects of

an arbitrary ﬁnite region R, behave as a state in

[[A

R

]].

Deﬁnition 15 (Generalised extended states).

The space [[A

G

]]

R

of generalised extended states

of A

G

is the topological dual of [[

¯

A

G

]]

QR

, i.e. the

Banach space [[

¯

A

G

]]

∗

QR

of bounded linear function-

als on [[

¯

A

G

]]

QR

, equipped with the norm

kρk

∗

:

= sup

kak

sup

=1

|(a|ρ)|. (29)

The operational norm on [[A

G

]]

R

, deﬁned as

kρk

op

:

= sup

a∈[[

¯

A

G

]]

Q

(2a − e

G

|ρ), (30)

coincides with the norm k·k

∗

, as we now prove.

Lemma 16. Let ρ ∈ [[A

G

]]

R

. Then

kρk

∗

= sup

a∈[[

¯

A

G

]]

Q

(2a − e

G

|ρ). (31)

Proof. Let 0 ≤ ε ≤ kρk

op

. We then have

0 ≤ kρk

op

− ε ≤ (2a − e

G

|ρ),

for some a ∈ [[

¯

A

G

]]

Q

. Invoking lemma 11, one has

e

G

+ (2a − e

G

) = 2a  0,

e

G

− (2a − e

G

) = 2(e

G

− a)  0,

and, by lemma 14, k2a − e

G

k

sup

≤ 1. Then it is

kρk

op

− ε ≤

(2a − e

G

|ρ)

k2a − e

G

k

sup

≤ kρk

∗

.

On the other hand, let us now pick a ∈ [[

¯

A

G

]]

R

with kak

sup

= 1. Then, by lemma 15, e

G

±a  0.

If we deﬁne a

±

:

=

1

2

(e

G

± a), we have a

±

 0,

and

a

+

+ a

−

= e

G

, (a

+

− a

−

) = a, (32)

which by lemma 12 implies that a

±

∈ [[

¯

A

G

]]

Q

.

Then

±(a|ρ) = (2a

±

− e

G

|ρ).

Thus, |(a|ρ)| ≤ kρk

op

, for every ε > 0, and taking

the supremum on the l.h.s. we obtain kρk

∗

≤

kρk

op

.

Technically speaking, the two norms k·k

sup

and

k·k

∗

are a base and order-unit norm pair [51].

What is missing now is the notion of a convex

set of proper states, the extended preparations of

our inﬁnite system A

G

. Let us then give the fol-

lowing deﬁnition.

Deﬁnition 16 (Local restriction). Given a state

ρ ∈ [[A

G

]]

R

, the local restriction of ρ to S ∈ R

(G)

is the functional ρ

|S

∈ [[A

S

]]

R

deﬁned through

(a|ρ

|S

)

:

= (a

S

|ρ), ∀a ∈ [[

¯

A

S

]]. (33)

The notion of a restriction can be brought fur-

ther, considering inﬁnite regions, by invoking the

isomorphism J

R

deﬁned in Eq. (25), as follows.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 19

Deﬁnition 17 (Restriction). Given a state ρ ∈

[[A

G

]]

R

, the restriction of ρ to S ∈ R

(G)

is the

functional ρ

|S

∈ [[A

S

]]

R

deﬁned through

(a|ρ

|S

)

:

= (J

−1†

S

a|ρ), ∀a ∈ [[

¯

A

S

]]. (34)

Deﬁning

ˆ

J

−1

S

by duality as

(a|

ˆ

J

−1

S

ρ)

:

= (J

−1†

S

a|ρ), (35)

we can then equivalently express Eq. (34) as

ρ

|S

=

ˆ

J

−1

S

ρ. (36)

Representing generalised extended states through

diagrams, and reminding eq. (28), the above

equations can be recast as

ρ

|S

S

a

=

ρ

R

J

−1

S

a

=

ρ

S

a

G\S

e

,

which can be taken as the deﬁnition of the equa-

tion

ρ

|S

S

:

=

ρ

S

G\S

e

. (37)

We can now introduce the set of states [[A

G

]]

as the special set of generalised extended states

whose restrictions are all states. In other words,

an element ρ of the space [[A

G

]]

R

is a state if,

restricted to any ﬁnite region R ∈ R

(G)

, it deﬁnes

a state for that region.

Deﬁnition 18 (Extended states). The set [[A

G

]]

of states in the space of generalised extended

states [[A

G

]]

R

is the set of those elements ρ ∈

[[A

G

]]

R

such that for every R ∈ R

(G)

the local re-

striction of ρ to R is a state ρ

|R

∈ [[A

R

]]. Deter-

ministic states, whose set is denoted by [[A

G

]]

1

,

are those states ρ ∈ [[A

G

]] with (e

G

|ρ) = 1.

One can easily verify that the set [[A

G

]] is con-

vex, as a consequence of convexity of [[A]] for ev-

ery ﬁnite system A. As we did for ﬁnite systems,

we also deﬁne a positive cone generated by [[A

G

]].

Moreover, it is easy to check that the restriciton

of a state to an inﬁnite region S ∈ R

(G)

is s state

in [[A

S

]].

Deﬁnition 19 (Extended positive cone). The set

[[A

G

]]

+

in the space of generalised extended states

[[A

G

]]

R

is the cone of those elements ρ ∈ [[A

G

]]

R

such that there exist ¯ρ ∈ [[A

G

]] and µ ≥ 0 such

that ρ = µ¯ρ.

We can now show an important result about

the restriction of states.

Lemma 17. The restriction

ˆ

J

−1

R

[[A

G

]] of the set

of states of A

G

coincides with [[A

R

]].

Proof. It is a straightforward exercise to verify

that

ˆ

J

−1

R

[[A

G

]] ⊆ [[A

R

]]. For the converse, let

ρ ∈ [[A

R

]], and let us now construct a state σ ∈

[[A

G

]] such that σ

|R

= ρ. We deﬁne

¯

R

:

= G \ R.

There are two possible situations:

¯

R ∈ R

(G)

, or

¯

R ∈ R

(G)

. If

¯

R ∈ R

(G)

, let a

T

∈ [[

¯

A

G

]]

LR

, with

T ∈ R

(G)

. We deﬁne

(a|σ)

:

= (a|ρ

|T ∩R

⊗ ν

|T ∩

¯

R

), (38)

for an arbitrary ν ∈ [[A

¯

R

]], where we introduced

the notation σ

|S

0

for S

0

⊆ S ∈ R

(G)

, and σ a state

of the composite system σ ∈ [[A

S

]] = [[A

S

0

A

S\S

0

]],

with

σ

|S

0

S

0

:

=

σ

S

0

S\S

0

e

.

Clearly, the functional σ is bounded, and thus

it can be extended to a functional on the full

space [[

¯

A

G

]]

QR

, i.e. σ ∈ [[A

G

]]

R

. Moreover one can

easily verify that σ

|T

∈ [[A

T

]] for every T ∈ R

(G)

,

thus σ is the desired state in [[A

G

]]. A similar

construction can be carried out for the case

¯

R ∈

R

(G)

, taking ν ∈ [[A

¯

R

]]. By construction, also in

this case σ ∈ [[A

G

]], as can be straightforwardly

checked.

The next result that we prove is that quasi-

local eﬀects a ∈ [[

¯

A

G

]]

Q

are positive on the set

[[A

G

]].

Lemma 18. Let a ∈ [[

¯

A

G

]]

Q

, and ρ ∈ [[A

G

]].

Then (a|ρ) ≥ 0

Proof. Let a ∈ [[

¯

A

G

]]

Q

be a local eﬀect a

R

=

[a

R

]. Then, by deﬁnition, (a

R

|ρ) = (a|ρ

|R

) ≥

0. Now, let a = [a

n

R

n

]. We have (a|ρ) =

lim

n→∞

(a

n

R

n

|ρ) ≥ 0.

The following result draws a ﬁrst analogy be-

tween the properties of the sup-norm for ﬁnite

systems and that for inﬁnite parallel composi-

tions.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 20

Lemma 19. Let ρ ∈ [[A

G

]]

+

. Then kρk

∗

=

(e

G

|ρ).

Proof. Let ρ ∈ [[A

G

]]

+

. Then, by virtue of

lemma 18, (a|ρ) ≥ 0 for all a ∈ [[

¯

A

G

]]. Let now

kak

sup

= 1. By lemma 15, one has e

G

± a  0,

and thus (e

G

|ρ) ≥ |(a|ρ)|. Then (e

G

|ρ) ≥ kρk

∗

.

On the other hand, being e

G

a legitimate el-

ement of [[

¯

A

G

]]

Q

with ke

G

k

sup

= 1, one has

kρk

∗

= sup

kak

sup

=1

|(a|ρ)| ≥ (e

G

|ρ).

Thanks to lemma 12, we have the following

corollary.

Corollary 9. Let ρ ∈ [[A

G

]]. Then 0 ≤ (a|ρ) ≤

(e

G

|ρ) for every a ∈ [[

¯

A

G

]]

Q

.

We can now show that also in the inﬁnite case

every state is proportional to a deterministic one.

Let us ﬁrst prove a preliminary lemma.

Lemma 20. Let ρ ∈ [[A

G

]]

R

. If (a|ρ) = 0 for

every a ∈ [[

¯

A

G

]]

Q

, then ρ = 0.

Proof. If (a|ρ) = 0 for every a ∈ [[

¯

A

G

]]

Q

then,

using lemma 11, we have

(2a − e

G

|ρ) = (a|ρ) − (e

G

− a|ρ) = 0, ∀a ∈ [[

¯

A

G

]]

Q

.

This implies that kρk

∗

= 0, and thus ρ = 0.

Proposition 7. Every state ρ ∈ [[A

G

]] is propor-

tional to a deterministic state ¯ρ ∈ [[A

G

]]

1

.

Proof. Let ρ ∈ [[A

G

]]. If ρ = 0, then for every

¯ρ ∈ [[A

G

]]

1

it is ρ = 0¯ρ. Let then ρ 6= 0. By

lemmas 18 and 20, there must exist a ∈ [[

¯

A

G

]]

such that (a|ρ) > 0. By corollary 9, one then

has (e

G

|ρ) ≥ (a|ρ) > 0. By deﬁnition, for every

S ∈ R

(G)

, we have

(e

A

S

|ρ

|S

) = (e

S

|ρ) = (e

G

|ρ) > 0.

Thus, if we set ¯ρ

:

= ρ/(e

G

|ρ), for every S ∈ R

(G)

we obtain

(e

A

S

|¯ρ

|S

) = (e

G

|¯ρ) = 1,

implying that ¯ρ

|S

∈ [[A

S

]]

1

for every S ∈ R

(G)

.

Then, ¯ρ ∈ [[A

G

]]

1

, and ¯ρ is such that ρ = (e

G

|ρ)¯ρ.

We now show that the operational norm is

equivalently deﬁned on [[

¯

A

G

]]

QR

by

kak

op

:

= sup

ρ∈[[A

G

]]

|(a|ρ)|.

Indeed, let a = [a

nR

n

]. Then

|(a|ρ)| = lim

n→∞

|(a

nR

n

|ρ)| = lim

n→∞

|(a

n

|ρ

|R

n

)|.

This implies that

|(a|ρ)| ≤ |(a

n

|ρ

|R

n

)| + ε ≤ ka

nR

n

k

op

+ ε,

thus sup

ρ∈[[A

G

]]

|(a|ρ)| ≤ kak

op

. On the other

hand, for every a

nR

n

and every ε, there exists

ρ

ε

∈ [[A

R

n

]] such that ka

nR

n

k

op

≤ |(a

n

|ρ

ε

)| + ε.

Now, by lemma 17, this implies that for every ε

there exists ˜ρ

ε

∈ [[A

G

]] such that

ka

nR

n

k

op

− ε ≤ |(a

nR

n

|˜ρ

ε

)|.

Finally, this implies that for every ε there exists

ρ

ε

∈ [[A

G

]] such that

kak

op

− 3ε ≤ ka

nR

n

k

op

− 2ε

≤ |(a

nR

n

|˜ρ

ε

)| − ε

≤ |(a|˜ρ

ε

)|. (39)

Thus, in conclusion, kak

op

≤ sup

ρ∈[[A

G

]]

|(a|ρ)|.

A crucial property of [[A

G

]] is that extended

states are separating for [[

¯

A

G

]]

Q

, namely if for ev-

ery state two generalised eﬀects give the same

value, then they coincide.

Theorem 6 (States separate eﬀects). Let a ∈

[[

¯

A

G

]]

QR

. If (a|ρ) = 0 for all ρ ∈ [[A

G

]]

1

, then

a = 0.

Proof. Let a = [a

n

R

n

] ∈ [[

¯

A

G

]]

QR

, and (a|ρ) = 0

for every ρ ∈ [[A

G

]]

1

. Then for every ρ ∈ [[A

G

]]

1

one has

|(a

n

|ρ)| = |(a

n

− a|ρ)| ≤ ka

n

− ak

sup

,

and thus for every ε > 0 there exists n

0

such

that, for n ≥ n

0

, ka

n

k

op

≤ ε. Thus, we have

kak

op

= 0. As a consequence, for every ε > 0

there exists n

0

such that for n ≥ n

0

one has

ka

n

k

sup

≤ kka

n

k

op

≤ ε,

and thus kak

sup

= 0. Now, this implies that a =

0.

Remark 2. The above result is made possible

by requirement (15) that the sup-norm and the

operational norm for eﬀects of ﬁnite systems A

are bounded as

∀a ∈ [[

¯

A]]

R

kak

sup

≤ kkak

op

,

with k independent of the system A. In par-

ticular, this is true under the assumption (16),

i.e. that [[

¯

A]]

∗

+

≡ [[A]]

+

.

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We can now prove the main consequence of

uniqueness of the deterministic eﬀect e

G

for A

G

,

i.e. that e

G

is the unique eﬀect that amounts to

1 on [[A

G

]]

1

.

Proposition 8. Let a ∈ [[

¯

A

G

]]

Q

. Then (a|ρ) = 1

for all ρ ∈ [[A

G

]]

1

if and only if a = e

G

.

Proof. Notice that, since e

G

 a for every a ∈

[[

¯

A

G

]]

Q

, and (b|ρ) ≥ 0 for every b ∈ [[

¯

A

G

]]

Q

and

ρ ∈ [[A

G

]]

1

, we have

0 ≤ (e

G

− a|ρ) = 1 − (a|ρ),

which implies (a|ρ) ≤ 1 for every ρ ∈ [[A

G

]]

1

.

Now, if (a|ρ) = 1 for every ρ ∈ [[A

G

]]

1

, we have

(e

G

− a|ρ) = 0, ∀ρ ∈ [[A

G

]]

1

,

which by theorem 6 implies a = e

G

.

A class of states of particular interest is that

of quasi-local states, which are intuitively under-

stood as states whose preparation can be arbi-

trarily approximated by local procedures. These

live in small subspaces of the space [[A

G

]]

R

. Their

construction is similar to that of quasi-local ef-

fects, but diﬀers form it in a relevant respect,

that is the necessity of deﬁning arbitrary refer-

ence states. The construction is inspired by pio-

neering works on inﬁnite tensor products by von

Neumann and Murray [49], and can be found in

appendix B.

4.3 Quasi-local transformations

Now we are going to deﬁne the quasi-local alge-

bra [[A

G

→ A

G

]]

QR

of transformations. In usual

approaches to quantum inﬁnite systems, one usu-

ally introduces the eﬀect algebra, that in the OPT

case would correspond to the space of quasi-local

eﬀects. However, one can easily understand that,

unlike eﬀects in a general OPT, the quantum ef-

fect algebra contains far more information than

the mere structure of the space of eﬀects. Indeed,

in the quantum eﬀect algebra one can ﬁnd every

operator on the Hilbert space, and in turn, the

operational role of a linear operator is to provide a

Kraus representation of a transformation. In this

perspective, one can also understand why mul-

tiplication of eﬀects has a meaning at all: there

is no point in multiplying eﬀects, but multiplica-

tion of Kraus operators is the mathematical rep-

resentation of sequential composition. Thus, the

algebraic structure of eﬀects conveniently sum-

marises information about every kind of event in

the theory: eﬀects, with their coarse-graining rep-

resented by the sum, and transformations, with

sequential composition represented by multiplica-

tion.

Cellular automata are deﬁned in quantum the-

ory by specifying their action on the eﬀect alge-

bra. Such an action thus provides information

about how both eﬀects and transformations are

transformed by the cellular automaton. For a

general OPT, however, such a compact algebraic

structure embodying all the relevant information

is absent, and the deﬁnition of a CA on the space

of eﬀects is not suﬃcient: we need to specify how

the CA transforms transformations.

For these reasons we construct now the Banach

algebra of quasi-local transformations, in a way

that is very closely reminiscent of the construc-

tion of quasi-local eﬀects. Here, in order to deﬁne

a local transformation, we recur to the notion of a

transformation that acts non-trivially on ﬁnitely

many systems, while it acts as the identity on the

remaining ones. In other words, the role that is

played by the deterministic eﬀect in the case of

local eﬀects is played here by the identity trans-

formation. Also in this case, the topological clo-

sure will be taken in the sup-norm, and this choice

is of great relevance to ensure a Banach algebra

structure for the closure.

Let us then introduce the Banach algebra of

quasi-local transformations.

Deﬁnition 20. Let R ⊆ G be an arbitrary ﬁnite

region of G. We deﬁne local transformation A

R

of A

G

any pair (A , R), where A ∈ [[A

R

→ A

R

]]

R

.

The action A

†

R

of A

R

on Pre[[

¯

A

G

]]

LR

is deﬁned as

follows

R\S

A

†

R

a

S

R∩S

S\R

:

=

R\S

A

R

R\S

e

R∩S R∩S

a

S

S\R

. (40)

We also deﬁne Pre[[A

G

→ A

G

]]

LR

as the set of

local transformations A

R

.

Lemma 21. Let a

S

∼ b

T

. Then A

†

R

a

S

∼ A

†

R

b

T

.

Proof. By hypothesis, there exists c ∈ [[

¯

A

S∩T

]]

R

such that

a = c ⊗ e

S\T

, b = c ⊗ e

T \S

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 22

Then we have

(A

†

R

a

S

| = (c ⊗ e

S\T

⊗ e

R\S

|(A ⊗ I

S\R

),

(A

†

R

b

T

| = (c ⊗ e

T \S

⊗ e

R\T

|(A ⊗ I

T \R

).

By Eq. (10) the thesis follows.

We can now deﬁne an equivalence relation be-

tween local transformations as follows

A

S

∼ B

T

⇔

(

A = C ⊗ I

S\T

,

B = C ⊗ I

T \S

,

(41)

for some C ∈ [[A

S∩T

]]

R

. The set of local transfor-

mations is then deﬁned as

[[A

G

→ A

G

]]

LR

:

= Pre[[A

G

→ A

G

]]

LR

/ ∼

Lemma 22. Let A

R

∈ Pre[[A

G

→ A

G

]]

LR

. Then

for every ﬁnite region H ∈ R

(G)

such that H ∩

R = ∅, one has (A ⊗ I

A

H

)

R∪H

∈ Pre[[A

G

→

A

G

]]

LR

, and (A ⊗ I

A

H

)

R∪H

∼ A

R

.

Proof. It is straightforward to verify that (A ⊗

I

A

H

)

R∪H

∼ A

R

by direct inspection of the deﬁn-

ing equation (41).

We can now provide a way to identify a canon-

ical representative of the equivalence class of A

R

,

which is deﬁned in analogy to the case of eﬀects

as follows.

Deﬁnition 21. The minimal representative of

the equivalence class A

R

, denoted as

˜

A

R

A

, is de-

ﬁned through

R

A

:

=

\

S∈R

(A ,R)

S,

˜

A

R

A

∼ A

R

, (42)

where R

(A ,R)

is the set of all those ﬁnite regions

S ⊆ G for which there exists B ∈ [[A

S

→ A

S

]]

R

such that B

S

∼ A

R

.

Lemma 23. The minimal representative exists

and is unique.

The proof follows step by step that of

lemma 73.

The ﬁrst result that we need to prove is the

following.

Lemma 24. Let A

R

∼ B

S

. Then for every ef-

fect a

T

∈ Pre[[

¯

A

G

]]

LR

, one has

A

†

R

a

T

∼ B

†

S

a

T

. (43)

Proof. If (

˜

C )

R

C

is the minimal representative of

A

R

∼ B

S

, by Eq. (41), one has

A =

˜

C ⊗ I

R\R

C

,

B =

˜

C ⊗ I

S\R

C

.

Now, by the deﬁning equation 40, we have that

(A

†

R

a

T

| = (˜a ⊗ e

(T ∪R)\R

a

|(

˜

C ⊗ I

(T ∪R)\R

C

)

=(˜a ⊗ e

[T ∪(R∩S)]\R

a

|(

˜

C ⊗ I

[T ∪(R∩S)]\R

C

)

⊗ (e

R\S

|,

(B

†

S

a

T

| = (˜a ⊗ e

(T ∪S)\R

a

|(

˜

C ⊗ I

(T ∪S)\R

C

)

=(˜a ⊗ e

[T ∪(R∩S)]\R

a

|(

˜

C ⊗ I

[T ∪(R∩S)]\R

C

)

⊗ (e

S\R

|,

and ﬁnally by Eq. (123) this implies the thesis.

By virtue of lemmas 21 and 24, we can deﬁne

the action of A

R

∈ [[A

G

→ A

G

]]

LR

on local eﬀects

[[

¯

A

G

]]

R

as

A

†

R

[a

T

]

:

= [A

†

R

a

T

],

where we leave the square braces to denote equiv-

alence classes for ∼ in Pre[[

¯

A

G

]]

LR

, for the sake of

clarity. We can now make local transformations

into a vector space as we did for local eﬀects, as

follows.

Deﬁnition 22. Let A

R

, B

S

∈ [[A

G

→ A

G

]]

LR

,

and h ∈ R. Then we deﬁne

hA

R

:

=

(

(hA )

R

h 6= 0,

0

∅

h = 0,

A

R

+ B

S

:

= C

R∪S

C

:

= A ⊗ I

S\R

+ B ⊗ I

R\S

.

Moreover, the following operation makes the

set of local transformations into an algebra.

Deﬁnition 23. Let A

R

, B

S

∈ [[A

G

→ A

G

]]

LR

.

Then we deﬁne

A

R

B

S

:

= ({A ⊗ I

S\R

}{B ⊗ I

R\S

})

R∪S

.

We omit the straightforward proof that the

above deﬁnition is well deﬁned, i.e. independent

of the choice of representatives in the classes of

A

R

and B

S

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 23

Deﬁnition 24. The algebra of generalized lo-

cal transformations is the unital algebra [[A

G

→

A

G

]]

LR

of ﬁnite real combinations of local trans-

formations, with unit 1

∅

and null element 0

∅

.

In order to close the algebra of local operations

we introduce the topology given by the sup-norm,

given in the following, in analogy to the case of

eﬀects. We also discuss the interplay of the sup-

norm topology with that given by the operational

norm, that we deﬁne right away.

Deﬁnition 25. The operational norm kA

R

k

op

of

˜

A

R

A

= A

R

∈ [[A → A]]

LR

is deﬁned by the

following expression

kA

R

k

op

:

= k

˜

A k

op

. (44)

Proposition 9. Let A

R

∈ [[A → A]]

LR

. Then

kA

R

k

op

= kA k

op

.

Proof. Let A

R

∼

˜

A

R

A

, and let R = R

A

∪ R

0

,

with R

A

∩ R

0

= ∅. Then by Eq. (41) we have

A =

˜

A ⊗ I

A

R

0

,

Now, by deﬁnition of k·k

op

(see [8]) it straightfor-

wardly follows that kB ⊗ I

C

k

op

= kBk

op

. Thus,

kA

R

k

op

= k

˜

A k

op

= kA k

op

.

Unfortunately, the operational norm does not

enjoy the basic property that would make the al-

gebra of transformations into a Banach algebra,

i.e. it is not true that kA Bk

op

≤ kA k

op

kBk

op

.

For this reason, in analogy with the case of quasi-

local eﬀects, we introduce a second norm, the sup-

norm, whose interplay with the operational norm

will make it possible to deﬁne the closed Banach

algebra of quasi-local transformations.

This is obtained extending the deﬁnition of

sup-norm to the algebra of local transformations

of A

G

.

Deﬁnition 26. The sup-norm kA

R

k

sup

of A

R

∈

[[A

G

→ A

G

]]

LR

is deﬁned as

kA

R

k

sup

:

= inf J(

˜

A ) = k

˜

A k

sup

. (45)

Proposition 10. Let A

R

∈ [[A

G

→ A

G

]]

LR

.

Then kA

R

k

sup

= kA k

sup

.

Proof. Let A

R

∼

˜

A

R

A

, and let R = R

A

∪ R

0

,

with R

A

∩ R

0

= ∅. Then by Eq. (41) we have

A =

˜

A ⊗ I

A

R

0

,

and by corollary 2, kA k

sup

= k

˜

A k

sup

=

kA

R

k

sup

.

Corollary 10. Let A , B ∈ [[A

G

→ A

G

]]

LR

.

Then kA Bk

sup

≤ kA k

sup

kBk

sup

.

Proof. The result is a straightforward conse-

quence of propositions 10 and 2.

Corollary 11. On [[A

G

→ A

G

]]

LR

, one has

kA

R

k

op

≤ kA

R

k

sup

.

Proof. The result is a straightforward conse-

quence of propositions 9 and 10, and corol-

lary 4.

The above results allow us to extend the lo-

cal algebra of events into the quasi-local alge-

bra, which is a Banach algebra. The construc-

tion is analogous to that of quasi-local states, and

proceeds as follows. First we deﬁne the algebra

[[A

G

→ A

G

]]

CR

of sup-Cauchy sequences of local

transformations, i.e. its elements are sequences

A : N → [[A

G

→ A

G

]]

LR

:: n 7→ A

n

R

n

such that

for every ε > 0 there exists n

0

∈ N such that

for all m, n ≥ n

0

, one has kA

n

− A

m

k

sup

< ε.

Now we deﬁne the equivalence relation between

elements of [[A

G

→ A

G

]]

CR

: A

∼

=

B if for every

ε > 0 there exists n

0

∈ N such that for every

n ≥ n

0

one has kA

n

− B

n

k

sup

< ε. Finally, we

deﬁne the quasi-local algebra as follows.

Deﬁnition 27. The space of quasi-local trans-

formations of A is deﬁned as

[[A

G

→ A

G

]]

QR

:

= [[A

G

→ A

G

]]

CR

/

∼

=

.

Deﬁnition 28. An element A in [[A

G

→ A

G

]]

QR

is an event if A = [A

nR

n

] for a sequence A

nR

n

such that A

n

∈ [[A

R

n

→ A

R

n

]] for all n ∈ N. The

set of events will be denoted by [[A

G

→ A

G

]]

Q

.

An element A ∈ [[A

G

→ A

G

]]

Q

is a channel

if A = [A

n

R

n

] for a sequence A

n

R

n

such that

A

n

∈ [[A

R

n

→ A

R

n

]]

1

for all n ∈ N. The set of

channels will be denoted by [[A

G

→ A

G

]]

Q1

.

The subset of [[A

G

→ A

G

]]

QR

containing ele-

ments A = [A

n

R

n

] for a sequence A

n

R

n

such

that A

n

∈ [[A

R

n

→ A

R

n

]]

+

for all n ∈ N will be

denoted by [[A

G

→ A

G

]]

Q+

. Equivalently, we will

write A  0 for A ∈ [[A

G

→ A

G

]]

Q+

In the following, in analogy with the ﬁnite case,

we will write A  B if A − B  0. The algebra

of quasi-local transformations [[A

G

→ A

G

]]

QR

is

deﬁned as follows.

Deﬁnition 29. Let A = [A

n

R

n

], B = [B

n

S

n

].

We deﬁne A B

:

= [A

n

R

n

B

n

S

n

].

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 24

The product A B is well deﬁned. Indeed:

1. A

n

R

n

B

n

S

n

is a Cauchy sequence.

2. For every A

0

n

R

0

n

, B

0

n

S

0

n

in the equivalence

classes deﬁning A and B, respectively, one

has A

n

R

n

B

n

S

n

∼

=

A

0

n

R

0

n

B

0

n

S

0

n

.

Item 1 can be easily proved since we have

kA

n

B

n

− A

m

B

m

k

sup

=kA

n

B

n

− A

m

B

n

+ A

m

B

n

− A

m

B

m

k

sup

≤kA

n

− A

m

k

sup

kB

n

k

sup

+ kA

m

k

sup

kB

n

− B

m

k

sup

.

Similarly, for item 2 we have

kA

n

B

n

− A

0

n

B

0

n

k

sup

=kA

n

B

n

− A

0

n

B

n

+ A

0

n

B

n

− A

0

n

B

0

n

k

sup

≤kA

n

− A

0

n

k

sup

kB

n

k

sup

+ kA

0

n

k

sup

kB

n

− B

0

n

k

sup

.

The quasi-local algebra is actually a real Ba-

nach algebra if equipped with the norm k·k

sup

, as

we now prove.

Proposition 11. The algebra [[A

G

→ A

G

]]

QR

equipped with the norm k·k

sup

is a Banach al-

gebra.

Proof. We only need to prove that kA Bk

sup

≤

kA k

sup

kBk

sup

. Let A = [A

n

R

n

] and B =

[B

n

S

n

], respectively. Now, for every A

n

R

n

and

B

n

S

n

one has

k(A

n

⊗ I

S

n

\R

n

)(B

n

⊗ I

R

n

\S

n

)k

sup

≤ kA

n

⊗ I

S

n

\R

n

k

sup

kB

n

⊗ I

R

n

\S

n

k

sup

,

and by proposition 10 kA Bk

sup

≤

kA k

sup

kBk

sup

.

The subsets [[A

G

→ A

G

]]

Q

and [[A

G

→ A

G

]]

Q1

are convex, and [[A

G

→ A

G

]]

Q+

is a convex cone.

Moreover, they are closed in the sup-norm.

Proposition 12. The sets [[A

G

→ A

G

]]

Q

,

[[A

G

→ A

G

]]

Q1

, and [[A

G

→ A

G

]]

Q+

are closed

in the sup-norm.

Proof. The argument is the same in both cases.

Let {A

m

}

m∈N

be a Cauchy sequence with

A

m

∈ [[A

G

→ A

G

]]

Q∗

for all m, with ∗ =

“nothing”, 1, +. By deﬁnition, A

m

= [A

mnR

mn

],

with A

mn

∈ [[A

R

mn

→ A

R

mn

]]

∗

. Then one eas-

ily proves that the sequence {A

mmR

mm

}

m∈N

is

Cauchy, and its limit is A

:

= lim

m→∞

A

mmR

mm

,

which by deﬁnition is in [[A

G

→ A

G

]]

Q∗

.

Proposition 13. Let A ∈ [[A

G

→ A

G

]]

QR

. Then

kA k

op

≤ kA k

sup

. (46)

Proof. By deﬁnition kA k

op

= lim

n→∞

kA

n

k

op

and kA k

sup

= lim

n→∞

kA

n

k

sup

, where

{A

n

R

n

}

n∈N

⊆ [[A

G

→ A

G

]]

LR

is a Cauchy

sequence converging to A , thus by proposition 3

one has

kA

n

k

op

≤ kA

n

k

sup

, ∀n ∈ N,

which implies the inequality in the limit.

Now, we can prove the following lemma.

Lemma 25. Let a ∈ [[

¯

A

G

]]

CR

be a Cauchy se-

quence, and let A ∈ [[A

G

→ A

G

]]

LR

. We then

have

A

†

a ∈ [[

¯

A

G

]]

CR

. (47)

Proof. We just need to evaluate

kA

†

a

m

− A

†

a

n

k

sup

, and using proposition 2 one

has

kA

†

(a

m

− a

n

)k

sup

≤ ka

m

− a

n

k

sup

kA k

sup

.

Since the sequence a is Cauchy, also A

†

a is

Cauchy.

Lemma 26. Let a, b ∈ [[

¯

A

G

]]

CR

be equivalent

Cauchy sequences, and let A ∈ [[A

G

→ A

G

]]

LR

.

We then have

[A

†

a

nR

n

] = [A

†

b

nS

n

]. (48)

Proof. We can bound kA

†

a

n

− A

†

b

n

k

sup

using

proposition 2, obtaining

kA

†

(a

n

− b

n

)k

sup

≤ ka

n

− b

n

k

sup

kA k

sup

.

Since [a

nR

n

] = [b

nS

n

], the thesis follows.

As a consequence of the above results, all local

transformations leave the space [[

¯

A

G

]]

QR

invari-

ant. We now prove a further important result:

the above statement can be extended to all the

quasi-local algebra.

Theorem 7. The quasi-local algebra [[A

G

→

A

G

]]

QR

leaves the space [[

¯

A

G

]]

QR

invariant, and

for every A ∈ [[A

G

→ A

G

]]

QR

and every a ∈

[[

¯

A

G

]]

QR

, it is

kA

†

ak

sup

≤ kak

sup

kA k

sup

. (49)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 25

Proof. By proposition 2 and lemma 25, the thesis

is true for A ∈ [[A

G

→ A

G

]]

LR

and a ∈ [[

¯

A

G

]]

LR

.

Let now a = [a

pR

p

] and A ∈ [[A

G

→ A

G

]]

LR

.

Then

kA

†

a

p

k

sup

≤ ka

p

k

sup

kA k

sup

,

and taking the limit for p → ∞ we obtain the

thesis. Finally, let A = [A

nR

n

]. In this case,

by lemma 25 A

†

n

a ∈ [[

¯

A

G

]]

QR

for every n ∈ N.

Moreover, {A

†

n

a}

n∈N

∈ [[

¯

A

G

]]

CR

. Indeed, by the

result we just proved

k(A

†

n

− A

†

m

)ak

sup

≤ kak

sup

kA

n

− A

m

k

sup

.

Finally, by lemma 25, we have

kA

†

n

ak

sup

≤ kak

sup

kA

n

k

sup

,

and taking the limit for n → ∞ we obtain the

thesis.

A remarkable result that will be important in

the following is given by the following lemma.

Lemma 27. Let A ∈ [[A

G

→ A

G

]]

Q

. Then for

every a ∈ [[

¯

A

G

]]

Q

one has A

†

a ∈ [[

¯

A

G

]]

Q

.

Proof. By hypothesis, A = [A

nR

n

] and a =

[a

mS

m

], with A

n

∈ [[A

R

n

→ A

R

n

]] and a

m

∈

[[

¯

A

S

m

]]. Thus A

†

n

a

n

∈ [[

¯

A

R

n

∪S

n

]]. Now, one can

easily verify that lim

n→∞

A

†

n

a

n

= A

†

a, which

proves the statement.

In particular, one has the following condition.

Lemma 28. Let A ∈ [[A

G

→ A

G

]]

Q+

. Then

A ∈ [[A

G

→ A

G

]]

Q

iﬀ A

†

e

G

∈ [[

¯

A

G

]]

Q

.

Proof. By deﬁnition, one has A ∈ [[A

G

→ A

G

]]

Q

if and only if A = [A

nR

n

] with A

n

∈ [[A

R

n

→

A

R

n

]], and by our assumptions the latter is equiv-

alent to a

n

:

= A

†

n

e

A

R

n

∈ [[

¯

A

R

n

]]. Now, since

ka

nR

n

− a

mR

m

k

sup

≤ kA

nR

n

− A

mR

m

k

sup

, one

has that A ∈ [[A

G

→ A

G

]]

Q

if and only if

A

†

e

G

= [a

nR

n

] ∈ [[

¯

A

G

]]

Q

.

Lemma 29. Let A ∈ [[A → B]]

Q+

, and a

:

=

A

†

e

G

. Then kA k

sup

= kak

sup

.

Proof. By deﬁnition, A = [A

nR

n

], with A

n

∈

[[A

R

n

→ A

R

n

]]

+

. Now, kA

n

R

n

k

sup

= kA

n

k

sup

,

and by lemma 7, kA

n

k

sup

= kA

†

n

e

R

n

k

sup

=

kA

†

nR

n

e

G

k

sup

. Then,

kA k

sup

= lim

n→∞

kA

†

n

R

n

e

G

k

sup

= kA

†

e

G

k

sup

.

On the same line, we can provide the following

condition for A ∈ [[A

G

→ A

G

]]

Q+

to be a channel.

Lemma 30. Let A ∈ [[A

G

→ A

G

]]

Q+

. Then

A ∈ [[A

G

→ A

G

]]

Q1

iﬀ A

†

e

G

= e

G

.

Proof. First, let A be a channel. By deﬁnition

it must be A = lim

n→∞

A

nR

n

, where A

nR

n

are

local channels. Thus, by theorem 3, A

†

nR

n

e

G

=

e

G

. It follows that A

†

e

G

= lim

n→∞

A

†

nR

n

e

G

=

e

G

. Now, for the converse, by deﬁnition A ∈

[[A

G

→ A

G

]]

Q+

iﬀ A = [A

n

R

n

] with A

n

R

n

∈

[[A

R

n

→ A

R

n

]]

+

for all n ∈ N. Let us take n ≥ n

0

so that kA

n

R

n

− A k

sup

≤ ε. Then we have

|kA

n

R

n

k

sup

− kA k

sup

| ≤ ε.

Now, by lemma 29, since A

†

e

G

= e

G

, we

have kA k

sup

= 1. Let us deﬁne A

0

n

R

n

:

=

A

n

R

n

/kA

n

R

n

k

sup

. Clearly,

kA

0

n

R

n

− A

n

R

n

k

sup

= kA

n

R

n

k

sup

|1 − 1/kA

n

R

n

k

sup

| ≤ ε.

This implies that A

0

n

R

n

is a sequence converg-

ing to A , with kA

0

n

R

n

k

sup

= 1. As a con-

sequence, there exist channels C

nR

n

such that

(C

n

− A

0

n

)

R

n

 0. Again by lemma 7,

k(C

n

− A

0

n

)

R

n

k

sup

= k(C

n

− A

0

n

)

†

R

n

e

R

n

k

sup

= ke

G

− A

0

n

†

R

n

e

G

k

sup

= k(A − A

0

nR

n

)

†

e

G

k

sup

≤ k(A − A

0

n

R

n

)k

sup

≤ ε.

This implies that A = [A

0

nR

n

] = [C

n

R

n

], and

then A ∈ [[A

G

→ A

G

]]

1

.

The following weak version of the converse of

lemma 28 can be easily proved.

Lemma 31. Let a ∈ [[

¯

A

G

]]

LR

. Then there exists

A ∈ [[A

G

]]

LR

such that

a = A

†

e

G

. (50)

Moreover, for a ∈ [[

¯

A

G

]]

L

, Eq. (50) is satisﬁed

with A ∈ [[A

G

→ A

G

]]

L

.

Proof. Let a = a

R

∈ [[

¯

A

G

]]

LR

. Let us deﬁne

A

R

:

= |σ)(a|, where σ ∈ [[A

R

]]

1

is an arbitrary

deterministic state. Then clearly

A

†

R

e

G

= a

R

.

It is also clear that, for a = a

R

∈ [[

¯

A

G

]]

L

,

the above deﬁned transformation A

R

is in

[[A

G

]]

L

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 26

One can take a further step, and prove the fol-

lowing result that will be of crucial importance in

the remainder.

Corollary 12. Let a ∈ [[

¯

A

G

]]

QR

. Then there

exists a sequence of quasi-local transformations

A

nR

n

such that lim

n→∞

A

†

nR

n

e

G

= a. If a ∈

[[

¯

A

G

]]

Q

the sequence can be found in [[A

G

→

A

G

]]

Q

.

Notice that the sequence A

n

R

n

is not necessar-

ily Cauchy, thus one cannot conclude that for ev-

ery a ∈ [[

¯

A

G

]]

QR

there is A ∈ [[A

G

→ A

G

]]

QR

such

that a = A

†

e

G

. However, the following lemma

completes the picture.

Proposition 14. Let a ∈ [[

¯

A

G

]]

QR

. Then there

exists A ∈ [[A

G

→ A

G

]]

QR

such that

a = A

†

e

G

. (51)

Moreover, for a ∈ [[

¯

A

G

]]

Q∗

with ∗ ∈

{“nothing”,1,+} one can ﬁnd the above A in the

set [[A

G

→ A

G

]]

Q∗

.

Proof. Let us deﬁne the sets

T

ε

a

:

= {A ∈ [[A

G

→ A

G

]]

QR

| kA

†

e

G

− ak

sup

≤ ε}.

The sets T

ε

a

are not empty for every a and every

ε > 0, as a consequence of corollary 12. More-

over, they are closed. Indeed, let {A

n

}

n∈N

be a

Cauchy sequence in T

ε

a

. This implies that by deﬁ-

nition a

n

:

= A

†

n

e

G

∈

¯

B

ε

(a). Since the ball

¯

B

ε

(a)

is closed, also lim

n→∞

a

n

= (lim

n→∞

A

n

)

†

e

G

∈

¯

B

ε

(a), namely lim

n→∞

A

n

∈ T

ε

a

. Finally, one

clearly has that for ε < δ it is T

ε

a

⊆ T

δ

a

, and the

diameter d

ε

:

= sup{kA − Bk

sup

| A , B ∈ T

ε

a

} is

a non-decreasing function of ε. Let then

T

0

a

:

=

\

ε>0

T

ε

a

.

Either lim

ε→0

+

d

ε

> 0, in which case T

0

a

is non

empty, and each of its elements satisﬁes A

†

e

g

=

a, or the limit is 0, in which case, by Cantor’s

intersection theorem for complete metric spaces,

T

0

a

= {A } is a singleton, with A

†

e

G

= a. The

same argument can be applied in each of the cases

where a ∈ [[

¯

A

G

]]

Q∗

, considering the closed sets

T

ε∗

a

:

= T

ε

a

∩ [[A

G

→ A

G

]]

Q∗

.

We now prove a result that is analogous to

lemma 12.

Lemma 32. Let A ∈ [[A

G

→ A

G

]]

Q+

, and

C − A ∈ [[

¯

A

G

→ A

G

]]

Q+

for some C ∈ [[A

G

→

A

G

]]

Q1

. Then A ∈ [[A

G

→ A

G

]]

Q

.

Proof. By deﬁnition, there are three sequences

A

nR

n

, A

0

nS

n

in [[A

G

→ A

G

]]

L

and C

nT

n

in [[A

G

→

A

G

]]

L1

such that lim

n→∞

A

n

= A , lim

n→∞

A

0

n

=

C − A , and lim

n→∞

C

n

= C . This implies that

kC

n

− (A

n

+ A

0

n

)k

sup

≤ ε,

and consequently there is a sequence of channels

D

n

in [[A

G

→ A

G

]]

L1

such that

εD

n

+ C

n

− (A

n

+ A

0

n

)  0.

By the above equation, it is easy to check that

F

n

−

1

1 + ε

A

n

 0, F

n

−

1

1 + ε

A

0

n

 0,

where F

n

is the channel ε/(1+ε)D

n

+1/(1+ε)C

n

.

Thus, by theorem 5, both A

n

/(1+ε) and A

0

n

/(1+

ε) belong to [[A

G

→ A

G

]]

L

. Let ε

m

be a real

Cauchy sequence converging to 0, and consider

the corresponding sequences B

m

:

= A

n

m

/(1 +

ε

m

), B

0

m

:

= A

0

n

m

/(1 + ε

m

). Since

kB

m

− A

n

m

k

sup

=



ε

m

1 + ε

m



kA

n

m

k

sup

,

kB

0

m

− A

0

n

m

k

sup

=



ε

m

1 + ε

m



kA

0

n

m

k

sup

,

one has A = [A

nR

n

] = [B

mR

m

] and C − A =

[A

0

nS

n

] = [B

0

mS

m

], with B

n

and B

0

n

beloning to

[[A

G

→ A

G

]]

L

. Thus, A , C − A ∈ [[A

G

→ A

G

]]

Q

.

We will now deﬁne the action of the quasi-local

algebra of transformations on quasi-local states.

The action is deﬁned by duality as follows.

Deﬁnition 30 (Dual action of the quasi-local

algebra). Let A ∈ [[A

G

→ A

G

]]

QR

. We deﬁne the

map

ˆ

A : [[A

G

]]

R

→ [[A

G

]]

R

as follows

(a|

ˆ

A ρ)

:

= (A

†

a|ρ), ∀a ∈ [[

¯

A

G

]]

QR

. (52)

The ﬁrst thing we need to prove is that the

map

ˆ

A : [[A

G

]]

R

→ [[A

G

]]

R

is linear and bounded.

Proposition 15. For every A ∈ [[A

G

→ A

G

]]

QR

the map

ˆ

A : [[A

G

]]

R

→ [[A

G

]]

R

is linear and

bounded.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 27

Proof. As to linearity, it is suﬃcient to consider

that

(a|

ˆ

A [xρ

1

+ yρ

2

]) = (A

†

a|xρ

1

+ yρ

2

)

= x(A

†

a|ρ

1

) + y(A

†

a|ρ

2

)

= (a|x

ˆ

A ρ

1

) + (a|y

ˆ

A ρ

2

),

which by deﬁnition implies

ˆ

A (xρ

1

+ yρ

2

) = x

ˆ

A ρ

1

+ y

ˆ

A ρ

2

.

In order to prove boundedness, it is suﬃcient to

consider Eq. (49), to conclude that

|(a|

ˆ

A ρ)| = |(A

†

a|ρ)| ≤ kA

†

ak

sup

kρk

∗

≤ kak

sup

kA k

sup

kρk

∗

,

which implies that

k

ˆ

A ρk

∗

≤ kA k

sup

kρk

∗

.

We now prove that [[A

G

]]

QR

is invariant under

the quasi-local algebra, and more precisely, every

space [[A

G

]]

(B)

ρ

0

QR

is invariant.

Theorem 8. Let A ∈ [[A

G

→ A

G

]]

QR

. Then

ˆ

A [[A

G

]]

QR

⊆ [[A

G

]]

QR

,

and for every (B, ρ

0

),

ˆ

A [[A

G

]]

(B)

ρ

0

QR

⊆ [[A

G

]]

(B)

ρ

0

QR

.

Proof. We start observing that, by Eq. (40),

when we apply A

†

R

a

S

to a local state τ

T

in

[[A

G

]]

(B)

ρ

0

QR

, we obtain

A

R

A

(R∩S)\T

A

S\(R∪T )

A

R\(S∪T )

A

R∩S∩T

A

(R∩T )\S

A

T \(R∪S)

A

(T ∩S)\R

ρ

0(R∪S)\T

τ

T

a

S

e

,

for every S and every a

S

∈ [[

¯

A

S

]]

R

, which implies

ˆ

A

R

τ

T

A

B(R\T )

A

R∩T

A

S\R

(53)

=

ρ

0B(R\T )

A

B(R\T )\(R\T )

A

R\T

A

R

A

R\T

τ

T

A

R∩T

A

R∩T

A

T \R

.

This means that local states in [[A

G

]]

(B)

ρ

0

QR

are

mapped to local states in the same sector un-

der

ˆ

A

R

. Now, by proposition 15, Cauchy se-

quences are mapped to Cauchy sequences, so that

ˆ

A

R

[[A

G

]]

(B)

ρ

0

QR

⊆ [[A

G

]]

(B)

ρ

0

QR

. Moreover, again by

proposition 15, also Cauchy sequences of local

transformations satisfy the same condition, thus

ˆ

A [[A

G

]]

(B)

ρ

0

QR

⊆ [[A

G

]]

(B)

ρ

0

QR

. Similarly, one can eas-

ily prove that

ˆ

A [[A

G

]]

QR

⊆ [[A

G

]]

QR

.

As a consequence of the above theorem, the ac-

tion of [[A → A]]

Q

on the space [[A

G

]]

QR

of quasi-

local states is decomposed into many irreducible

representations, one for every space [[A

G

]]

(B)

ρ

0

Q

.

Remark 3. Notice that, given A , B ∈ [[A

G

→

A

G

]]

QR

, one has

(A B)

†

= B

†

A

†

,

\

(A B) =

ˆ

A

ˆ

B. (54)

Let R ∈ R

(G)

, and let A ∈ [[A

G

→ A

G

]]

Q

.

Suppose that A = [A

mR

m

] and for every m ∈ N,

R∩R

m

= ∅. This implies that, for S

m

= R

m

∪R,

A = [A

0

mS

m

], with A

0

mS

m

= (A

m

⊗ I

R

)

S

m

, for

A

0

m

∈ [[A

S

m

\R

]]

R

. We then write A = B ⊗ I

R

.

Notice that in this case, given ρ ∈ [[A

G

]]

R

, one has

kρ

|R

− (

ˆ

A ρ)

|R

k

op

= k(

ˆ

A

m

ρ)

|R

− (

ˆ

A ρ)

|R

k

op

≤ k(

ˆ

A

m

−

ˆ

A )ρk

∗

≤ εkρk

∗

,

and then (

ˆ

A ρ)

|R

= ρ

|R

.

For R ∈ R

(G)

, there is a straightforward

ordered Banach algebra isomorphism between

[[A

R

→ A

R

]]

QR

and [[A

(G)

R

→ A

(G)

R

]]

QR

, where the

latter is deﬁned as

[[A

(G)

R

→ A

(G)

R

]]

QR

:

= {A

R

| A ∈ [[A

R

→ A

R

]]

R

}.

A similar isomorphism can be found also in the

case of inﬁnite regions R ∈ R

(G)

, where by deﬁ-

nition [[A

(G)

R

→ A

(G)

R

]]

QR

is the closed subalgebra

of Cauchy classes [A

n

R

n

] with R

n

⊆ R for every

n ∈ N. The isomorphism in this case is given by

[[A

R

→ A

R

]]

†

QR

= J

†

R

[[A

(G)

R

→ A

(G)

R

]]

†

QR

J

−1

R

†

,

(55)

where J

†

R

: [[

¯

A

(G)

R

]]

QR

→ [[

¯

A

R

]]

QR

is deﬁned in

Eq. (25). In the following, when we want to spec-

ify that a given quasi-local transformation A is

in the subalgebra [[A

R

→ A

R

]]

QR

, we will write

A

R

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 28

Remark 4. Notice that every quasi-local trans-

formation in [[A

R

→ A

R

]]

QR

does not only rep-

resent a linear map on [[

¯

A

R

]]

QR

, but a family of

maps that represent the same transformation act-

ing on [[

¯

A

G

]] for every R ⊆ G. This is due to the

isomorphism of [[A

R

→ A

R

]]

QR

with the subalge-

bra [[A

(G)

R

→ A

(G)

R

]]

QR

given in Eq. (55).

The following result shows that [[A

(G)

R

→

A

(G)

R

]]

QR

preserves the subspace [[

¯

A

(G)

R

]]

QR

.

Lemma 33. Let A ∈ [[A

(G)

R

→ A

(G)

R

]]

QR

. Then

A

†

[[

¯

A

(G)

R

]]

QR

⊆ [[

¯

A

(G)

R

]]

QR

.

Proof. Let A

nR

n

be a sequence in the class of

A , with R

n

⊆ R for every n. Let a

mS

m

be a

sequence in the class of a ∈ [[

¯

A

(G)

R

]]

QR

, with S

m

⊆

R for all m. Then A

†

a = [(A

†

n

a

n

)

R

n

∪S

n

], and

for every n, R

n

∪ S

n

⊆ R. This implies that

A

†

a ∈ [[

¯

A

(G)

R

]]

QR

.

The following intuitive result states that an ef-

fect that is localised in the region R ⊆ G is ob-

tained by a quasi-local transformation that is lo-

calised in the same region. This has a straight-

forward proof that will be omitted.

Lemma 34. Let a ∈ [[

¯

A

(G)

R

]]

Q∗

, with ∗ ∈

{“nothing”, 1, +, R}. Then

a = A

†

e

G

,

with A ∈ [[A

(G)

R

→ A

(G)

R

]]

Q∗

.

We conclude the section with a result that con-

ﬁrms, under restrictive hypotheses, an obvious

expectation.

Lemma 35. Let A ∈ [[A

G

→ A

G

]]

QR

. Then

kA k

sup

= inf J(A ), where

J(A )

:

= {λ ∈ R | ∃C ∈ [[A

G

→ A

G

]]

Q1

, λC ± A  0}.

Proof. Let λ ∈ J(A ). Then there exists a quasi-

local channel C such that λC ± A  0. By deﬁ-

nition, this implies that λC ± A = [B

±

nR

±

n

] with

B

±

 0. Deﬁning A

n

:

= 1/2(B

+

n

− B

−

n

) and

C

n

:

= 1/(2λ)(B

+

n

+ B

−

n

), by construction one

has

[C

nS

+

n

] = C , [A

nS

−

n

] = A ,

where the domains S

±

n

are suitably deﬁned. On

the other hand, C = [D

nT

n

] for D

nT

n

∈ [[A

T

n

→

A

T

n

]]

1

. Thus, one has kC

n

− D

n

k

sup

≤ ε, taking

suitable care in deﬁning the domain U

n

:

= S

+

n

∪

T

n

of C

n

− D

n

. In other terms, one can ﬁnd a

channel F

n

such that

εF

n

+ D

n

− C

n

 0,

εF

n

− D

n

+ C

n

 0.

In turn, this implies that

(1 + ε)e

U

n

− c

n

 0,

(ε − 1)e

U

n

+ c

n

 0,

where c

n

:

= C

†

n

e

U

n

. Considering the ﬁrst of the

above relations, we can deﬁne

G

n

:

=

1

1 + ε

C

n

+ |ρ

n

)(d

n

|,

where d

n

:

= e

U

n

− 1/(1 + ε)c

n

 0 and ρ

n

is an

arbitrary state in [[A

U

n

]]

1

. By our assumptions,

since G

n

 0 and G

†

n

e

U

n

= e

U

n

, G

n

is a channel.

Moreover, by construction we have that

λ(1 + ε)G

n

± A

n

= B

±

n

+ λ(1 + ε)|ρ

n

)(d

n

|  0,

thus λ(1 + ε) ∈ J(A

n

), i.e. λ(1 + ε) ≥ kA

n

k

sup

.

In the limit, we then have λ ≥ kA k

sup

. On the

other hand, if λ 6∈ J(A ), for every C ∈ [[A

G

→

A

G

]]

Q1

one has

λC + A 6 0 ∨ λC − A 6 0. (56)

Since the sequence λC ±A

n

converges to λC ±A

for every sequence A

n

converging to A , and the

positive cone is closed, for every C and every A

n

as above, there must be n

0

such that for n ≥ n

0

λC + A

n

6 0 ∨ λC − A

n

6 0. (57)

This holds, in particular, for every local C

n

with

the same domain as A

n

, which implies λ 6∈

J(A

n

). As a consequence, λ < kA

n

k

sup

, and

ﬁnally λ ≤ kA k

sup

. It is now easy to conclude

that inf J(A ) = kA k

sup

.

We now prove that the inﬁmum deﬁning the

sup norm is actually a minimum, analogously to

the case of eﬀects. The proof in this case is less

straightforward, and we ﬁrst need the following

lemma.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 29

Lemma 36. Given A ∈ [[A

G

→ A

G

]]

QR

, let

S

λ

A

⊆ [[A

G

→ A

G

]]

Q1

be the set of channels C

such that λC ± A  0. Then S

λ

A

is closed in the

sup-norm.

Proof. The empty set is closed, so we will con-

sider the case where S

λ

A

is not empty. Let

{C

n

}

n∈N

be a Cauchy sequence in S

λ

A

. Then

D

±

n

:

= λC

n

± A  0 are both Cauchy sequences,

that by deﬁnition converge to λC ± A  0. By

proposition 12, C ∈ [[A

G

→ A

G

]]

Q1

.

Proposition 16. Let A ∈ [[A

G

→ A

G

]]

QR

. Then

there exists a quasi local channel C such that

kA k

sup

C ± A  0. (58)

Proof. By lemma 35, for every ε > 0 there exists

C

ε

such that (kA k

sup

+ε)C

ε

±A  0. Let us then

consider the non empty sets S

ε

:

= S

kA k

sup

+ε

A

, that

are closed by lemma 36. It is straightforward to

verify that for δ < ε it is S

δ

⊆ S

ε

. Let d

ε

be the

diameter of S

ε

, namely d

ε

:

= sup{kC

1

− C

2

k

sup

|

C

1

, C

2

∈ S

ε

}. Notice that 0 ≤ d

ε

≤ 2. Being

d

ε

a non-decreasing function of ε, d

0

= lim

ε→0

+

exists. The limit d

0

is the diameter of

S

0

:

=

\

ε>0

S

ε

.

If d

0

> 0, then S

0

is clearly not empty, and each of

its elements is a channel C such that kA k

sup

C ±

A  0. If d

0

= 0, then by Cantor’s intersection

theorem for complete metric spaces the set S

0

is

a singleton S

0

= {C

∞

}, with kA k

sup

C

∞

± A 

0.

We ﬁnally prove a crucial result, that is the

converse of lemma 32. This shows that every

quasi-local event is an element of a quasi-local

test.

Proposition 17. Let A ∈ [[A

G

→ A

G

]]

Q+

.

Then A ∈ [[A

G

→ A

G

]]

Q

if and only if there

exists C ∈ [[A

G

→ A

G

]]

Q1

such that C − A ∈

[[A

G

→ A

G

]]

Q+

.

Proof. Suﬃciency is proved in lemma 32. Let us

then prove that if A ∈ [[A

G

→ A

G

]]

Q

then there

exists C ∈ [[A

G

→ A

G

]]

Q1

such that C − A  0.

First of all, notice that A = [A

nR

n

] with A

n

∈

[[A

R

n

→ A

R

n

]], and then by

kA k

sup

= lim

n→∞

kA

n

k

sup

= lim

n→∞

kA

†

nR

n

e

G

k

sup

≤ 1.

By proposition 16, there exists C ∈ [[A

G

→

A

G

]]

Q1

such that

C − A  kA k

sup

C − A  0.

5 Global update rules

We have now set the theoretical background

needed for the deﬁnition of a cellular automaton.

In the present section we give the deﬁnition of

an update rule (UR), that along with the system

A

G

—the array of cells, whose deﬁnition has been

analyzed in detail in the previous sections—will

provide the backbone of the notion of a cellu-

lar automaton. We then introduce the notion of

a global update rule (GUR), as a family of up-

date rules satisfying suitable admissibility condi-

tions in order to represent a local action when

extended to composite systems A

G

C. A GUR

thus describes the evolution occurring when we

apply the global update rule on a joint state of

A

G

and C, leaving C unaﬀected. We then analyse

the main features of GURs, and in particular we

focus on the causal inﬂuence relation given by a

global update rule, and the block decomposition

that allows one to calculate the action of a GUR

on local eﬀects using only local transformations.

Remark 5. In the remainder we will assume

that for every g ∈ G the system A

g

is not trivial,

namely A

g

6' I.

First of all let V

†

be a bounded automorphism

of the space [[

¯

A

G

]]

QR

of quasi-local eﬀects. Then,

with a notation that is reminiscent of the one

adopted for quasi-local transformations, we will

denote its action as

V

†

: [[

¯

A

G

]]

QR

→ [[

¯

A

G

]]

QR

:: a 7→ V

†

a.

The action on eﬀects allows us to deﬁne the dual

map that acts on the space of extended states, as

follows

(a|

ˆ

V ρ)

:

= (V

†

a|ρ), ∀a ∈ [[

¯

A

G

]]

QR

. (59)

One can easily verify that

ˆ

V is linear and

bounded, thanks to boundedness of the map V

†

.

Unlike the usual notion of a map in quantum the-

ory or theories with local discriminability, how-

ever, the action of

ˆ

V or V

†

is not suﬃcient to

identify uniquely the corresponding transforma-

tion. In particular, those actions are not suﬃ-

cient to determine that of V ⊗ I

C

, when sys-

tem A

G

is considered as a part of the composite

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 30

system A

G

C. For this reason we need to deﬁne

an automorphic transformation of [[

¯

A

G

]]

QR

with

care. One further delicate issue is the follow-

ing. When one conjugates a quasi-local transfor-

mation A with an automorphism V

†

, obtianing

A

0

= V

−1†

A V

†

, the latter is a linear bounded

map on [[

¯

A

G

]]

QR

, but there is no guarantee that

it is still an element of the quasi-local algebra

[[A

G

→ A

G

]]

QR

, which is surely a desideratum for

candidate cellular automata. In the remainder,

when we write A

G

C we mean to deal with a sys-

tem A

G

0

with G

0

= G ∪ h

0

and A

h

0

∼

=

C. In

view of the above observations, we introduce the

following notion

Deﬁnition 31 (Automorphic family). An auto-

morphic family of maps on [[

¯

A

G

]]

QR

is a collec-

tion of automorphisms V

†

C

of [[

¯

A

G

¯

C]]

QR

, one for

every C ∈ Sys(Θ), with the properties i) for every

A ∈ [[A

C

→ A

C

]]

R

one has V

−1†

C

A

†

h

0

V

†

C

= A

†

h

0

;

ii) for ﬁxed C

0

and every choice of system C

1

,

setting C = C

0

C

1

and A

G

0

:

= A

G

C

0

, and for ev-

ery A ∈ [[A

G

0

→ A

G

0

]]

QR

, there exist A

0

, A

00

∈

[[A

G

0

→ A

G

0

]]

QR

such that

V

−1†

C

A

†

G

0

V

†

C

= A

0†

G

0

, (60)

A

†

G

0

= V

†

C

A

00†

G

0

V

−1†

C

, (61)

where A

0†

= (V

−1†

C

0

A

†

V

†

C

0

), and A

00†

=

(V

−1†

C

0

A

†

V

†

C

0

)

We remark that, as a special case of item ii)

in deﬁnition 31, taking C

0

≡ I and C = C

1

, for

every choice of system C, and every A ∈ [[A

G

]]

QR

,

one has

V

−1†

C

A

†

G

V

†

C

= (V

−1†

I

A

†

G

V

†

I

)

G

,

V

†

C

A

†

G

V

−1†

C

= (V

†

I

A

†

G

V

−1†

I

)

G

.

When referring to an automorphic family we

will use the shorthand V

†

. One has to remind,

however, that this symbol refers to a family of

automorphisms. In particular, the deﬁnition is

meant to allow for formulas such as V A V

−1

,

V [[A

G

→ A

G

]]

$∗

V

−1

, for $ = Q, L and ∗ =

R, +, 1 or nothing. The meaning of the two ex-

pressions above is given by the following deﬁni-

tions:

V A V

−1

:

= {

ˆ

V

C

0

C

(

ˆ

A ⊗

ˆ

I

C

)

ˆ

V

−1

C

0

C

| C ∈ Sys(Θ)},

(62)

V [[A

G

0

→ A

G

0

]]

$∗

V

−1

:

=

{V A V

−1

| A ∈ [[A

G

0

→ A

G

0

]]

$∗

}. (63)

Notice that by deﬁnition of an automorphic fam-

ily and by Eq. (62), one has

V A V

−1

= {(

ˆ

V

C

0

ˆ

A

ˆ

V

−1

C

0

) ⊗

ˆ

I

C

| C ∈ Sys(Θ)},

(64)

namely an automorphic family maps [[A

G

0

→

A

G

0

]]

QR

onto itself surjectively. In particular, this

is true of [[A

G

→ A

G

]]

QR

. This implies that for

A ∈ [[A

G

C → A

G

C]]

QR

, V A V

−1

represents a

quasi-local transformation in [[A

G

C → A

G

C]]

QR

.

An automorphic family is deﬁned as to satisfy

some necessary conditions that we require for the

representatives of a transformation of the form

V ⊗ I

C

. However, the deﬁnition of an automor-

phic family is not suﬃcient to ensure consistency.

This fact must be taken into account in the next

subsection. Suﬃcient conditions will be only pro-

vided through the notion of admissibility.

5.1 Update rule

We can now use the notion of automorphic family

to introduce update rules. An update rule is de-

ﬁned as a family of automorphisms, one for each

extended system A

G

C, with the constraint that

the rule must act reversibly on the set of states

and, by conjugation, on the cone of positive local

transformations.

Deﬁnition 32 (Update rule). An update rule

(UR) is a triple (G, A, V

†

), where V

†

is an au-

tomorphic family of isometric maps on [[

¯

A

G

]]

QR

such that

1. the maps

ˆ

V

C

leave [[A

G

C]] invariant, i.e.

ˆ

V

C

[[A

G

C]] = [[A

G

C]]; (65)

2. the map V · V

−1

leaves the cones [[A

G

C →

A

G

C]]

L+

invariant, i.e. for every A ∈

[[A

G

C → A

G

C]]

L+

there are A

0

, A

00

∈

[[A

G

C → A

G

C]]

L+

such that

V A V

−1

= A

0

, (66)

A = V A

00

V

−1

. (67)

The ﬁrst result is that the inverse of a UR is a

UR.

Lemma 37. Let (G, A, V

†

) be a UR. Then

(G, A, V

−1†

) is a UR.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 31

Proof. In the ﬁrst place, the family V

−1

C

is an au-

tomorphic family: items i) and ii) in deﬁnition 31

are trivially proved. Now, if kV

†

C

ak

sup

= kak

sup

for every a ∈ [[

¯

A

G

¯

C]]

QR

, then kV

−1†

C

ak

sup

=

kV

†

C

V

−1†

C

ak

sup

= kak

sup

, which proves isometric-

ity of V

−1†

C

. From the deﬁnition in Eq. (59) one

can easily verify that

ˆ

(V

−1

C

) = (

ˆ

V

C

)

−1

. Now, it

is a straightforward observation that by Eq. (65),

ˆ

V

−1

C

[[A

G

C]] = [[A

G

C]]. Finally, multiplying both

sides of Eq. (66) by V

−1

to the left and by V to

the right, for every A ∈ [[A

G

C → A

G

C]]

L+

one

has

A = V

−1

A

0

V ,

which expresses the same condition as in Eq. (67)

for V

−1

, and similarly, multiplying both sides of

Eq. (67) by V

−1

to the left and by V to the right,

one obtains the same condition as in Eq. (66) for

V

−1

.

In the following we will often use V

†

to de-

note a UR, when G and A

G

are clear from the

context. The symbols

ˆ

V and V

†

will be often

used instead of

ˆ

V

C

and V

†

C

, respectively. Accord-

ingly, the symbols a, b, . . . and A , B, . . . will de-

note a ⊗ e

C

, b ⊗ e

C

, . . . and A ⊗ I

C

, B ⊗ I

C

, . . .,

respectively. Finally, we will sometimes write G

0

for G ∪ h

0

with A

h

0

∼

=

C, so that A

G

0

= A

G

C.

The next result states that an update rule

leaves the cone [[A

(G∪h

0

)

G

→ A

(G∪h

0

)

G

]]

L+

invariant.

Lemma 38. Let (G, A, V

†

) be a UR, and deﬁne

A

G

0

:

= A

G

C for arbitrary C ∈ Sys(Θ). Then the

map V · V

−1

leaves the cone [[A

(G

0

)

G

→ A

(G

0

)

G

]]

L+

invariant, namely for A ∈ [[A

(G

0

)

G

→ A

(G

0

)

G

]]

L+

Eqs. (66) and (67) hold with A

0

, A

00

∈ [[A

(G

0

)

G

→

A

(G

0

)

G

]]

L+

.

Proof. Let A ∈ [[A

G

0

G

→ A

G

0

G

]]

L+

⊆ [[A

G

0

→

A

G

0

]]

L+

. By deﬁnition of UR

V A

G

V

−1

= A

0

G

∈ [[A

G

0

→ A

G

0

]]

L+

.

Moreover, by deﬁnition of an automorphic family

of maps and Eq. (64), we have

A

0

∈ [[A

(G

0

)

G

→ A

(G

0

)

G

]]

LR

.

The same holds exchanging V

−1

and V .

As a consequence of isometricity on [[

¯

A

G

¯

C]]

QR

,

a UR V

†

has a dual

ˆ

V that acts isometrically on

[[A

G

C]]

R

.

Lemma 39. Let V

†

be a UR. The maps

ˆ

V

C

on

[[A

G

0

]]

R

are isometric.

Proof. Let ρ ∈ [[A

G

C]]

R

, and remind that

kρk

∗

= sup

kak

sup

=1

|(a|ρ)|.

Then we have

k

ˆ

V

C

ρk

∗

= sup

kak

sup

=1

|(a|

ˆ

V

C

ρ)| = sup

kak

sup

=1

|(V

†

C

a|ρ)|

= sup

ka

0

k

sup

=1

|(a

0

|ρ)| = kρk

∗

,

thanks to isometricity and invertibility of V

†

C

.

Corollary 13. Let V

†

be a UR. Then

ˆ

V

C

[[A

G

C]]

1

= [[A

G

C]]

1

Proof. By eqs. (65) and (29) and isometricity of

ˆ

V

C

, one has

ˆ

V

C

[[A

G

C]]

1

⊆ [[A

G

C]]

1

. The same is

true for

ˆ

V

−1

C

, thus

ˆ

V

C

[[A

G

C]]

1

= [[A

G

C]]

1

.

The next results will allow us to conclude that

a UR preserves [[

¯

A

G

]]

Q

. We start proving that an

update rule V

†

preserves the unique deterministic

eﬀects e

G

0

.

Lemma 40. If V

†

is a UR, then V

†

e

G

0

= e

G

0

,

for A

G

0

:

= A

G

C and arbitrary C ∈ Sys(Θ).

Proof. By deﬁnition of a UR, for every ρ ∈ [[A

G

0

]]

one has

ˆ

V ρ ∈ [[A

G

0

]], and lemma 19, gives for

every ρ ∈ [[A

G

0

]]

(V

†

e

G

0

|ρ) = (e

G

0

|

ˆ

V ρ) = k

ˆ

V ρk

∗

= kρk

∗

= (e

G

0

|ρ).

Thanks to theorem 6, we then have V

†

e

G

0

= e

G

0

.

Corollary 14. Let V

†

be a UR. Then V

†

a

h

0

=

a

h

0

for all a

h

0

∈ [[A

(G

0

)

h

0

→ A

(G

0

)

h

0

]]

QR

.

Proof. By proposition 14, a

h

0

= A

†

h

0

e

G

0

. Thus,

by deﬁnition 31 and lemmas 40 and 56, we have

V

†

a

h

0

= V

†

A

†

h

0

V

−1†

e

G

0

= A

†

h

0

e

G

0

= a

h

0

.

We now prove that the cones of eﬀects [[

¯

A

G

0

]]

L+

,

[[

¯

A

G

]]

L+

, [[

¯

A

G

0

]]

Q+

and [[

¯

A

G

]]

Q+

are invariant un-

der the action of any UR.

Lemma 41. Let V

†

be a UR. Then

V

†

[[

¯

A

G

0

]]

L+

= [[

¯

A

G

0

]]

L+

, and V

†

[[

¯

A

G

0

]]

Q+

=

[[

¯

A

G

0

]]

Q+

. In particular, for C = I it is

V

†

[[

¯

A

G

]]

Q+

= [[

¯

A

G

]]

Q+

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 32

Proof. Let a ∈ [[

¯

A

G

0

]]

L+

. By proposition 14,

there exists A ∈ [[A

G

0

→ A

G

0

]]

L+

such that

a = A

†

e

G

0

. Then

V

†

a = V

†

A

†

e

G

0

= V

†

A

†

V

−1†

V

†

e

G

0

= V

†

A

†

V

−1†

e

G

0

= A

0†

e

G

0

= a

0

.

We observe that, by deﬁnition of UR and by

lemma 38, V · V

−1

preserves the cones [[A

G

0

→

A

G

0

]]

L+

. Thus, a

0

∈ [[

¯

A

G

0

]]

L+

, and V

†

[[

¯

A

G

0

]]

L+

⊆

[[

¯

A

G

0

]]

L+

. Since also V

−1†

is a UR, we have

V

†

[[

¯

A

G

0

]]

L+

= [[

¯

A

G

0

]]

L+

. Now, isometricity grants

that Cauchy sequences of local eﬀects in [[

¯

A

G

0

]]

L+

are mapped to Cauchy sequences in [[

¯

A

G

0

]]

L+

,

and that lim

n→∞

V

†

a

n

= V

†

lim

n→∞

a

n

. Fi-

nally, by proposition 12, V

†

[[

¯

A

G

0

]]

Q+

⊆ [[

¯

A

G

0

]]

Q+

.

Since also V

−1†

is a UR, we conclude that

V

†

[[

¯

A

G

0

]]

Q+

= [[

¯

A

G

0

]]

Q+

.

Proposition 18. Let V

†

be a UR. Then

V

†

[[

¯

A

G

0

]]

Q

= [[

¯

A

G

0

]]

Q

. In particular, for C = I,

V

†

[[

¯

A

G

]]

Q

= [[

¯

A

G

]]

Q

Proof. Let a ∈ [[

¯

A

G

0

]]

Q

. Then e

G

0

− a ∈ [[

¯

A

G

0

]]

Q+

,

by lemma 12, and thus, by virtue of lemmas 40

and 41, V

†

a, e

G

0

− V

†

a ∈ [[

¯

A

G

0

]]

Q+

. Again by

lemma 12, this implies that V

†

a ∈ [[

¯

A

G

0

]]

Q

, and

thus V

†

[[

¯

A

G

0

]]

Q

⊆ [[

¯

A

G

0

]]

Q

. Finally, since also

V

−1†

is a UR, the thesis follows.

Let now V

†

be an update rule. By our def-

inition, the map V · V

−1

preserves the cones

[[A

G

0

→ A

G

0

]]

L+

. We will now prove that V · V

−1

also preserves the cones [[A

G

0

→ A

G

0

]]

Q+

, as well

as the sets of transformations [[A

G

0

→ A

G

0

]]

L

and

[[A

G

0

→ A

G

0

]]

Q

.

We start with the following lemma

Lemma 42. Let V

†

be a UR. Then V · V

−1

leaves the sets [[A

G

0

→ A

G

0

]]

L1

invariant, namely

for A ∈ [[A

G

0

→ A

G

0

]]

L1

Eqs. (66) and (67) hold,

with A

0

, A

00

∈ [[A

G

0

→ A

G

0

]]

L1

. In particular,

for C = I the thesis holds with G

0

= G.

Proof. Let C ∈ [[A

G

0

→ A

G

0

]]

L1

. By deﬁni-

tion and by lemma 38, V C V

−1

= C

0

, and

C = V C

00

V

−1

, for C

0

, C

00

∈ [[A

G

0

→ A

G

0

]]

L+

.

By lemmas 37 and 40,

C

0†

e

G

0

= V

†

C

†

V

−1†

e

G

0

= e

G

0

,

C

00†

e

G

0

= V

−1†

C

†

V

†

e

G

0

= e

G

0

,

and due to lemma 30, C

0

, C

00

∈ [[A

G

0

→ A

G

0

]]

L1

.

Given a UR V

†

, the above result allows us to

prove that V ·V

−1

maps local transformations to

local transformations, and is isometric on [[A

G

0

→

A

G

0

]]

LR

.

Lemma 43. Let V

†

be a UR, and A ∈ [[A

G

0

→

A

G

0

]]

LR

. Then V [[A

G

0

→ A

G

0

]]

LR

V

−1

= [[A

G

0

→

A

G

0

]]

LR

, and

kV A V

−1

k

sup

= kA k

sup

. (68)

Proof. By proposition 16, there exists C ∈

[[A

G

0

→ A

G

0

]]

L1

such that

kA k

sup

C ± A  0.

By deﬁnition of a UR, we then have

kA k

sup

V C V

−1

± V A V

−1

 0, (69)

and being V C V

−1

∈ [[A

G

0

→ A

G

0

]]

L1

⊆

[[A

G

0

→ A

G

0

]]

LR

, it follows that V A V

−1

∈

[[A

G

0

→ A

G

0

]]

LR

. Consequently, we have

V [[A

G

0

→ A

G

0

]]

LR

V

−1

⊆ [[A

G

0

→ A

G

0

]]

LR

.

Finally, since V

−1†

is a UR, we have that

V [[A

G

0

→ A

G

0

]]

LR

V

−1

= [[A

G

0

→ A

G

0

]]

LR

.

Moreover, by lemma 42, Eq. (69) implies that

kA k

sup

∈ J(V A V

−1

). Thus kV A V

−1

k

sup

≤

kA k

sup

. Now, being V

−1†

a UR, the equality

kV A V

−1

k

sup

= kA k

sup

follows.

Lemma 44. Let V

†

be a UR. Then V [[A

G

0

→

A

G

0

]]

QR

V

−1

= [[A

G

0

→ A

G

0

]]

QR

, and

lim

n→∞

V A

n

V

−1

= V ( lim

n→∞

A

n

)V

−1

. (70)

Proof. Since V · V

−1

is isometric and surjective

on the dense submanifolds [[A

G

0

→ A

G

0

]]

LR

, by

lemma 43 it maps Cauchy sequences to Cauchy

sequences, and their equivalence classes to equiv-

alence classes, thus it maps [[A

G

0

→ A

G

0

]]

QR

to itself. Surjectivity follows from surjectivity

on [[A

G

0

→ A

G

0

]]

LR

. Eq. (70) can be indepen-

dently checked applying both sides to arbitrary

a ∈ [[

¯

A

G

0

]]

QR

.

Lemma 45. Let V

†

be a UR. Then V [[A

G

0

→

A

G

0

]]

Q1

V

−1

= [[A

G

0

→ A

G

0

]]

Q1

.

Proof. Let {C

n

}

n∈N

be a Cauchy sequence in

[[A

G

0

→ A

G

0

]]

L1

. Then by deﬁnition of UR

and lemma 44 we have that {V C

n

V

−1

}

n∈N

is a

Cauchy sequence in [[A

G

0

→ A

G

0

]]

L1

, and its limit

V C V

−1

is in [[A

G

0

→ A

G

0

]]

Q1

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 33

The last results ﬁnally allow us to prove that

for a UR V

†

, the map V ·V

−1

preserves the cones

[[A

G

0

→ A

G

0

]]

Q+

.

Lemma 46. Let V

†

be a UR. Then V [[A

G

0

→

A

G

0

]]

Q+

V

−1

= [[A

G

0

→ A

G

0

]]

Q+

.

Proof. By lemmas 38 and 44, along with the deﬁ-

nition of a UR, V ·V

−1

maps Cauchy sequences in

[[A

G

0

→ A

G

0

]]

L+

to Cauchy sequences in [[A

G

0

→

A

G

0

]]

L+

, thus V [[A

G

0

→ A

G

0

]]

Q+

V

−1

⊆ [[A

G

0

→

A

G

0

]]

Q+

. Since V

−1†

is a UR, the thesis fol-

lows.

Next, we show that for a UR V

†

, V

†

[[

¯

A

G

0

]]

LR

=

[[

¯

A

G

0

]]

LR

.

Lemma 47. Let V

†

be a UR. Then

V

†

[[

¯

A

G

0

]]

LR

= [[

¯

A

G

0

]]

LR

.

Proof. Let a

R

∈ [[

¯

A

G

0

]]

LR

. Then by proposi-

tion 14

a

R

= A

†

R

e

G

0

,

for some A

R

∈ [[A

G

0

→ A

G

0

]]

LR

. Now, we have

V

†

a

R

= V

†

A

†

R

e

G

0

= V

†

A

†

R

V

−1†

e

G

0

,

and by lemma 43 V

†

A

†

R

V

−1†

= A

0†

R

0

with A

0

R

0

∈

[[A

G

0

→ A

G

0

]]

LR

. Then V

†

a

R

= a

0

R

0

, with R

0

∈

R

(G

0

)

. As to surjectivity, it can be straightfor-

wardly proved exploiting lemma 37.

Finally, for a UR V

†

, the map V ·V

−1

preserves

the set of quasi-local transformations.

Lemma 48. Let V

†

be a UR. Then one has

V [[A

G

0

→ A

G

0

]]

Q

V

−1

= [[A

G

0

→ A

G

0

]]

Q

, (71)

V [[A

G

0

→ A

G

0

]]

L

V

−1

= [[A

G

0

→ A

G

0

]]

L

. (72)

Proof. Let A ∈ [[A

G

0

→ A

G

0

]]

Q

. Then kA k

sup

≤

1, and by proposition 17, there is C ∈ [[A

G

0

→

A

G

0

]]

Q1

such that

C − A  0.

By lemma 46, V A V

−1

∈ [[A

G

0

→ A

G

0

]]

Q+

.

Moreover, by lemma 42,V C V

−1

∈ [[A

G

0

→

A

G

0

]]

Q1

. By the above observations, we have

V A V

−1

 0,

V C V

−1

− V A V

−1

 0.

Finally, by lemma 32, one has

V A V

−1

, V C V

−1

− V A V

−1

∈ [[A

G

0

→ A

G

0

]]

Q

.

The same argument holds for A ∈ [[A

G

0

→

A

G

0

]]

L

. Since V

−1†

is a UR, surjectivity on both

sets is proved.

Clearly, given a UR V

†

, the image of the alge-

bra of local transformations of A

g

under V · V

−1

,

i.e.

V [[A

(G

0

)

g

→ A

(G

0

)

g

]]

QR

V

−1

⊆ [[A

(G

0

)

G

→ A

(G

0

)

G

]]

QR

,

is a subalgebra of [[A

(G

0

)

G

→ A

(G

0

)

G

]]

QR

isomorphic

to [[A

(G

0

)

g

→ A

(G

0

)

g

]]

QR

.

5.2 Admissibility and local action

As we have seen, independently of the nature of

system C, the map V · V

−1

does more than map-

ping linear maps to linear maps: it sends transfor-

mations to transformations. This is an important

result, as the deﬁnitions of automorphic family

and UR are meant to embody necessary condi-

tions for a family of maps to represent V ⊗ I

C

.

However, URs are not yet consistent with such

a factorisation. The reason for this is that lo-

cal discriminability in our theory is not assumed.

Thus, while knowledge of the linear maps V

†

C

is

suﬃcient to determine the action of the UR on

factorised eﬀects a

G

⊗ b

C

and, by conjugation, on

generalised transformations A

G

⊗ I

C

, and such

actions are consistent with a factorised action of

the form V ⊗ I

C

, still we do not have enough in-

formation to determine the action of V

†

C

on those

elements of [[

¯

A

G

¯

C]]

QR

that lie outside the sub-

space [[

¯

A

G

]]

QR

⊗ [[

¯

C]]

QR

(and analogously for the

conjugate action on transformations). For this

purpose we introduce in this section the notion

of admissibility.

We remind that G

0

stands for G ∪ h

0

, with

A

h

0

∼

=

C for some system C. Let R ∈ R

(G

0

)

be

a ﬁnite region. Since [[

¯

A

(G

0

)

R

]]

QR

is isomorphic to

[[

¯

A

R

]]

R

, whose size is D

R

:

= D

A

R

, we can ﬁnd a

basis {b

jR

}

D

R

j=1

⊆ [[

¯

A

(G

0

)

R

]]

QR

. Thus, by lemma 47,

V

†

[[

¯

A

(G

0

)

R

]]

QR

⊆ [[

¯

A

(G

0

)

R

0

]]

QR

for some ﬁnite region

R

0

∈ R

(G

0

)

, that can be obtained as

R

0

:

=

D

R

[

j=1

R

0

j

, V

†

b

jR

∈ [[

¯

A

(G

0

)

R

0

j

]]

QR

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 34

This implies that, for every ﬁnite region R ∈

R

(G

0

)

, the UR V

†

induces a linear map V (R)

†

:

[[

¯

A

R

]]

R

→ [[

¯

A

R

0

]]

R

. The following deﬁnition of ad-

missible UR is now given in terms of the proper-

ties of the induced maps V (R)

†

.

Deﬁnition 33. Let (G, A, V

†

) be a UR, and

let V (R)

†

be the induced maps on ﬁnite regions

R ⊆ G

0

, for arbitrary choice of A

h

0

∼

=

C. We call

the UR admissible if for every ﬁnite S ∈ R

(G)

the

family of maps V (S ∪ h

0

)

†

represents a transfor-

mation V

S

∈ [[A

S

0

→ A

S

]]

R

, i.e.

V (S ∪ h

0

)

†

= V

†

S

⊗ I

†

C

. (73)

Consider now a partition G = R ∪

¯

R, where

¯

R

:

= G \ R. We will write A

G

= A

R

⊗ A

¯

R

.

By deﬁnition, the space [[

¯

A

G

]]

QR

of quasi-local ef-

fects contains the spaces [[

¯

A

(G)

R

]]

QR

and [[

¯

A

(G)

¯

R

]]

QR

as closed subspaces. However, if the theory does

not satisfy local discriminability, it is not true

in general that [[

¯

A

G

]]

QR

= [[

¯

A

(G)

R

]]

QR

⊗ [[

¯

A

(G)

¯

R

]]

QR

.

This makes it diﬃcult to deﬁne a UR of the form

V ⊗ W .

In the remainder of the section we close this

gap. For this purpose, we start deﬁning what it

means, for a UR V

†

on A

G

, to act locally on R—

in symbols V

†

= V

0†

⊗ I

†

¯

R

. For our purpose,

we then need to provide a suitable deﬁnition of

the identity V

†

= V

0†

⊗ I

†

¯

R

, which satisﬁes the

necessary (but not suﬃcient) condition

(V

0†

⊗ I

†

¯

R

)(a

S

⊗ b

T

) = (V

0†

a

S

) ⊗ (b

T

),

for every S, T ∈ R

(G)

such that S ⊆ R and T ⊆

¯

R.

Deﬁnition 34. Let V

†

be a UR for A

G

. We

say that V

†

acts locally on the region R ⊆ G

if for every C and every ﬁnite region S ∈ R

(G

0

)

with G

0

= G ∪ h

0

and A

h

0

∼

=

C, the induced map

V (S)

†

has the form

V

†

(S) = W

†

S∩R

⊗ I

†

S\R

. (74)

As a consequence of the above deﬁnition we

now prove the following result, for which we do

not provide the proof, that is straightforward.

Lemma 49. Let (G, A, V

†

) be a UR acting lo-

cally on R. Then

V

†

(a

S

⊗ b

T

) = (W

†

S

a

S

) ⊗ b

T

, (75)

for S ⊆ R and T ⊆

¯

R. In particular,

V

†

[[

¯

A

(G

0

)

R

]]

QR

= [[

¯

A

(G

0

)

R

]]

QR

, and V

†

b = b for every

b ∈ [[

¯

A

(G

0

)

¯

R

]]

QR

.

Lemma 50. Let (G, A, V

†

) be a UR act-

ing locally on R. Then there exists a UR

(R, A, V

0†

) such that V

†

J

−1

R

0

†

a = J

−1†

R

0

V

0†

a

and V

−1†

J

−1†

R

0

a = J

−1†

R

0

V

0−1†

a for every a ∈

[[

¯

A

R

0

]]

QR

, with A

R

0

:

= A

R

C.

Proof. Linearity and isometricity of the restric-

tion V

†

R

0

of V

†

to [[

¯

A

(G

0

)

R

0

]]

QR

are straightforward

by lemma 49. Thus, also V

0†

:

= J

†

R

0

V

†

R

0

J

−1†

R

0

is linear and isometric. Let now A ∈ [[A

R

0

→

A

R

0

]]

Q

. Then one has

V

0†

A

†

V

0−1†

= J

†

R

0

V

†

R

0

J

−1†

R

0

A

†

J

†

R

0

V

−1†

R

0

J

−1†

R

0

= J

†

R

0

V

†

R

0

A

†

R

0

V

−1†

R

0

J

−1†

R

0

= J

†

R

0

(V

†

A

†

R

0

V

−1†

)

R

0

J

−1†

R

0

,

where A

†

R

0

:

= J

−1†

R

0

A

†

J

†

R

0

. In the above chain

of equalities we used the fact that, since V

†

,

V

−1†

and A

†

R

0

preserve the space [[

¯

A

(G

0

)

R

0

]]

QR

, then

also V

†

A

†

R

0

V

−1†

does, and one can thus deﬁne

its restriction (V

†

A

†

R

0

V

−1†

)

R

0

. Finally, since

V

†

is a UR, one has V

†

A

†

R

0

V

−1†

= A

0†

with

A

0

∈ [[A

(G

0

)

R

0

→ A

(G

0

)

R

0

]]

Q

. Thus, V

0−1

A V

0

∈

[[A

R

0

→ A

R

0

]]

Q

. A similar argument leads to

the existence of A

00

∈ [[A

R

0

→ A

R

0

]]

Q

such that

A = V

0−1

A

00

V

0

. Now, let ρ ∈ [[A

G

]]. Then ρ

deﬁnes a state ρ

|R

0

on [[

¯

A

R

0

]]

QR

, by lemma 17,

with

(a|ρ

|R

0

) = (a

R

0

|ρ) = (J

−1†

R

0

a|ρ),

for every a

R

0

∈ [[

¯

A

(G

0

)

R

0

]]

QR

. Now, we have

(a|[

ˆ

V ρ]

|R

0

) = (J

−1†

R

0

a|

ˆ

V ρ) = (V

†

J

−1†

R

0

a|ρ)

= (V

†

R

0

J

−1†

R

0

a|ρ) = (J

−1†

R

0

J

†

R

0

V

†

R

0

J

−1†

R

0

a|ρ)

= (J

−1†

R

0

V

0†

a|ρ) = (a|

ˆ

V

0

ρ

|R

0

).

We then showed that [

ˆ

V ρ]

|R

0

=

ˆ

V

0

(ρ

|R

0

). More-

over, since V preserves the set of states, [

ˆ

V ρ]

|R

0

is a state. Thus,

ˆ

V

0

[[A

R

0

]] ⊆ [[A

R

0

]]. The same

argument for V

0−1

brings to the conclusion that

ˆ

V

0

[[A

R

0

]] = [[A

R

0

]]. Thus, V

0†

is a UR for A

R

.

Moreover

V

†

J

−1†

R

0

a = V

†

R

0

J

−1†

R

0

a = J

−1†

R

0

V

0†

a,

and the same holds for V

−1

.

In the following, under the hypothesis of

lemma 50, we will write V

†

= V

0†

⊗ I

†

¯

R

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 35

Lemma 51. Let V

†

be a UR for [[

¯

A

G

]]

R

that acts

locally on R, and W

†

a UR that acts locally on

¯

R. Then V

0†

⊗ W

0†

:

= V

†

W

†

= W

†

V

†

.

We can now deﬁne a global update rule as fol-

lows.

Deﬁnition 35 (Global update rule). Let

(G, A, V

†

) be a UR. We call (G, A, V

†

) a Global

Update Rule (GUR), or global rule for short, if

(G, A, V

†

) and (G, A, V

−1†

) are admissible.

Remark 6. The set of GURs (G, A, V

†

) is a

group. The proof of closure under composi-

tion and associativity are tedious but straightfor-

ward, and we omit them here. Moreover, given

(G, A, V

†

) and (F, A, W

†

) one can deﬁne their

parallel composition as (G ∪ F, A, V

†

⊗ W

†

).

The analysis of properties of parallel composition

is beyond the scope of the present work, and will

be the subject of further studies.

We now provide an important example of

GUR. Let G = H ×{0, 1}, and let H

i

:

= (H, i) for

i = 0, 1. Let a

R

∈ [[

¯

A

(G

0

)

R

]]

QR

with A

h

0

∼

=

C. Let

R ∩ G = (R

0

, 0) ∪ (R

1

, 1). Then we deﬁne

˜

R

:

=

R

0

∪ R

1

⊆ H, and

¯

R

:

=

˜

R × {0, 1} ∪ R. Clearly,

R ⊆

¯

R. If (h, i) ∈

¯

R, then also (h, i⊕1) ∈

¯

R. Let

then ¯a

¯

R

:

= a

R

⊗ e

¯

R\R

.

Now, for every T ∈

R

(H)

let us deﬁne the GUR

(G, A, S

†

T

) by setting

S

†

T

[a

R

]

:

=

h

S

†

˜

R∩T

¯a



¯

R

i

, (76)

where S

†

X

:

=

N

h∈X

S

†

h

for X ∈ R

(H)

, and S

h

is the map that swaps A

(h,0)

and A

(h,1)

. It is

straightforward to verify that, for every T ∈ R

(H)

,

S

T

is a UR. Being deﬁned in terms of induced

maps, it is immediate to verify that S

†

T

admissi-

ble. Moreover, S

T

is involutive, i.e. S

−1

T

= S

T

.

Of particular interest for the following is the GUR

(G, A, S

†

H

).

Notice that, for a ﬁnite disjoint partition H =

∪

k

i=1

S

k

, we have S

†

H

=

N

k

i=1

S

†

S

k

. Thus, by

lemma 51, for every GUR (G, A, V

†

), and for ev-

ery i, j, we have

V (S

S

i

∪S

j

⊗ I

H\(S

i

∪S

j

)

)V

−1

= V (S

S

i

⊗ I

H\S

i

)V

−1

V (S

S

j

⊗ I

H\S

j

)V

−1

= V (S

S

j

⊗ I

H\S

j

)V

−1

V (S

S

i

⊗ I

H\S

i

)V

−1

,

where I

S

denotes I

(S,0)

⊗ I

(S,1)

.

For A ∈ [[A

R

→ A

R

]], it is easily veriﬁed that

A

(R,1)

= S

H

(A

(R,0)

)S

H

= S

R

(A ⊗ I

R

)S

R

,

A

(R,0)

= S

H

(A

(R,1)

)S

H

= S

R

(I

R

⊗ A )S

R

.

(77)

Finally, let us consider a GUR (G, A, U

†

⊗ V

†

).

One can prove that S

H

(U ⊗ V )S

H

= V ⊗ U ,

by the following argument. Let us consider a ∈

[[

¯

A

(G

0

)

R

]]

QR

. Then

S

†

H

(U ⊗ V )

†

S

†

H

a

R

= S

†

H

(U ⊗ V )

†

(S

†

˜

R

a)

¯

R

= S

†

H

[(U (

˜

R, 0)

†

⊗ V (

˜

R, 1)

†

)(S

†

˜

R

a)]

¯

R

= [S

†

˜

R

(U (

˜

R, 0)

†

⊗ V (

˜

R, 1)

†

)(S

†

˜

R

a)]

¯

R

= [V (

˜

R, 0)

†

⊗ U (

˜

R, 1)

†

a]

¯

R

= (V ⊗ U )

†

a

R

.

5.3 Causal inﬂuence

In the present subsection we deﬁne the notion of

causal inﬂuence that establishes, given a GUR,

whether the evolution allows interventions on one

system to aﬀect other systems after one step, or

more generally how many steps are needed for

such an inﬂuence to occur. We will often use the

notation [[A

(G)

R

C → A

(G)

R

C]]

∗∗

to denote [[A

(G

0

)

R

→

A

(G

0

)

R

]]

∗∗

, with G

0

= G ∪ h

0

and A

h

0

∼

=

C. In the

following we also adopt the convention that for

R ∈ R

(G)

,

¯

R denotes the complementary region

¯

R

:

= G \ R. Notice that, consistently with the

notation R = g, we will write ¯g to denote G \ g.

Deﬁnition 36 (Causal inﬂuence). We say that,

according to the global rule V , system g does not

causally inﬂuence g

0

if for every external system

C

V [[A

(G)

g

C → A

(G)

g

C]]

QR

V

−1

⊆ [[A

(G)

¯g

0

C → A

(G)

¯g

0

C]]

QR

.

On the other hand, we will say that according

to the global rule V system g causally inﬂuences

system g

0

, and write g  g

0

, in the opposite case,

i.e. when

∃C, F ∈ [[A

(G)

g

C → A

(G)

g

C]]

QR

:

(V ⊗ I

C

)F (V

−1

⊗ I

C

) 6∈ [[A

(G)

¯g

0

C → A

(G)

¯g

0

C]]

QR

.

The above deﬁnition gets a clear meaning as a

diagram: according to V there is causal inﬂuence

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 36

from g to g

0

if there exists F ∈ [[A

g

C → A

g

C]]

such that

C

F

C

g

V

−1

V

g

¯g ¯g

6=

C

F

0

C

¯g

0

¯g

0

g

0

g

0

,

(78)

and even better in the form

C

F

C

g

V

g

¯g ¯g

6=

C

F

0

C

¯g

0

V

¯g

0

g

0

g

0

.

Deﬁnition 37. We deﬁne forward and backward

g-neighbourhoods the sets N

+

g

:

= {g

0

|g  g

0

}

and N

−

g

:

= {g

0

|g

0

 g}, respectively.

In terms of diagrams, the forward g-

neighbourhood A

N

+

g

of g can be understood as

the smallest region K =

N

k∈K

A

g

0

k

such that for

every k ∈ K there exists C and F ∈ [[A

(G)

g

C →

A

(G)

g

C]]

QR

such that

C

F

C

g

V

g

¯g ¯g

(79)

=

C

F

0

C

N

+

g

V

N

+

g

¯

N

+

g

¯

N

+

g

.

with F

0

acting non trivially on g

0

k

, namely F

0

6∈

[[A

(G)

¯g

0

k

C → A

(G)

¯g

0

k

C]]

QR

. Notice that g may or may

not belong to N

+

g

.

Analogously to the deﬁnition of N

±

g

, one can

give a deﬁnition of N

±

R

for every ﬁnite region R ⊆

G.

Deﬁnition 38. We say that the system g

0

belongs

to the forward neighbourhood of R, and write g

0

∈

N

+

R

, if

∃C, F ∈ [[A

(G)

R

C → A

(G)

R

C]]

QR

:

(V ⊗ I

C

)F (V

−1

⊗ I

C

) 6∈ [[A

(G)

¯g

0

C → A

(G)

¯g

0

C]]

QR

.

The backward neighbourhood N

−

R

of R is deﬁned

as

N

−

R

:

= {g

0

∈ G | N

+

g

0

∩ R 6= ∅}. (80)

Clearly, one has

N

+

R

⊇

[

g∈R

N

+

g

. (81)

We will prove that the latter is actually an equal-

ity in the next section.

One might deﬁne causal inﬂuence without in-

voking any external system C, as follows.

Deﬁnition 39 (Individual inﬂuence). We say

that the system g does not individually inﬂuence

g

0

if

V [[A

(G)

g

→ A

(G)

g

]]

QR

V

−1

⊆ [[A

(G)

¯g

0

→ A

(G)

¯g

0

]]

QR

.

On the other hand, we will say that according to

the global rule V system g individually inﬂuences

system g

0

, and write g 

g

0

, in the opposite

case, i.e. when

∃F ∈ [A

(G)

g

→ A

(G)

g

]

QR

:

V F V

−1

6∈ [[A

(G)

¯g

0

→ A

(G)

¯g

0

]]

QR

.

The individual forward neighbourhood of g is

N

+

g

(g)

:

= {g

0

∈ G | g 

g

0

}.

A second deﬁnition that can be given without

referring to external systems C is the following.

Deﬁnition 40 (Cooperative inﬂuence). We say

that the region S does not cooperatively inﬂuence

g

0

if

V [[A

(G)

S

→ A

(G)

S

]]

QR

V

−1

⊆ [[A

(G)

¯g

0

→ A

(G)

¯g

0

]]

QR

.

In the opposite case we will write S 

S

g

0

. The

cooperative forward neighbourhood of a region S

is deﬁned as

N

+

S

(S)

:

= {g

0

∈ G | S 

S

g

0

}.

Notice now that in a theory without local dis-

criminability it may happen that a system g does

not individually inﬂuence g

0

but it does cooper-

atively, namely one can have g

0

∈ N

+

S

(S), but

g

0

6∈ N

+

g

(g) for any g ∈ S. In other words,

N

+

S

(S) ⊃

S

h∈S

N

+

h

(h). We call this phenomenon

non-local activation of causal inﬂuence. If such

a phenomenon occurs, however, it can be hard

to establish which of the “cooperating” systems

h ∈ S is responsible for causal inﬂuence on

g

0

∈ N

+

S

(S). This is the reason why we prefer

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 37

the former deﬁnition of causal inﬂuence, which

provides also this piece of information.

Given a GUR (G, A, V

†

), we can now construct

a graph Γ(G, E) with vertices g ∈ G and edges

E

:

= {(g, g

0

) ∈ G × G|g  g

0

}. We call such a

graph the graph of causal inﬂuence of the GUR

V

†

. In the next section we will prove that, if we

denote by N

±∗

g

the neighbourhoods for the GUR

V

−1

, the following relations hold

N

±

f

= N

∓∗

f

. (82)

We also deﬁne the local perspective present P

+

R

of R ∈ R

(G)

as the set P

+

R

:

= N

−

N

+

R

⊇ R,

namely the set of systems that can causally in-

ﬂuence the one-step future N

+

R

of R. Similarly

one can deﬁne the local retrospective present of R

as P

−

R

:

= N

+

N

−

R

⊇ R, namely the set of systems

that can be inﬂuenced by the one-step past N

−

R

of R. In general, P

+

R

6= P

−

R

. We will also need to

refer to the sets P

+∗

R

:

= N

+∗

N

+

R

.

5.3.1 Relation with signalling

The term inﬂuence that we used above is pre-

cisely deﬁned in Eq. (78). In the present sub-

section we will make a clear point that this def-

inition is stronger than the notion of signalling

that is customary in the literature on quantum

information theory [52, 53, 54] or in general op-

erational probabilistic theories [15]. In that con-

text, a channel C ∈ [[AB → A

0

B

0

]] is called non-

signalling (or semicausal) from B to A

0

if there

exists F ∈ [[A → A

0

]] such that

A

C

A

0

B B

0

e

=

A

F

A

0

B

e

. (83)

We will now show that, under the hypothesis that

N

+∗

g

is ﬁnite for every g ∈ G, the condition of

Eq. 79, in a suitably deﬁned sense, implies that

of Eq. (83). Let us consider the state ρ ∈ [[A

G

]]

R

.

Then (a

S

|

ˆ

A

T

ρ)

:

= (a

0

S∪T

|ρ) = (A

†

T

a

S

|ρ). In the

case where a

S

= e

G

, (A

†

T

e

G

|ρ) = (b

T

|ρ), where

b = A

†

e

T

. Let us now consider (

ˆ

V ρ)

|R

, which

by deﬁnition is the state in [[A

R

]]

R

such that, for

every a ∈ [[

¯

A

R

]]

R

(a|[

ˆ

V ρ]

|R

) = (J

R

†

a|

ˆ

V ρ) = (a

R

|

ˆ

V ρ).

Now, by lemma 31, one has

(a

R

|

ˆ

V ρ) = (e

G

|

ˆ

A

R

ˆ

V ρ)

= (e

G

|

ˆ

V

ˆ

V

−1

ˆ

A

R

ˆ

V ρ)

= (e

G

|

ˆ

V

ˆ

A

0

N

+∗

R

ρ)

= (e

G

|

ˆ

A

0

N

+∗

R

ρ)

= (b

N

+∗

R

|ρ) = (b|ρ

|N

+∗

R

).

In order to determine [

ˆ

V ρ]

|R

it is suﬃcient to

know ρ

|N

+∗

R

. Indeed, the linear map V

R

:

[[A

N

+∗

R

]]

R

→ [[A

R

]]

R

gives

(

ˆ

V ρ)

|R

= V

R

(ρ

|N

+∗

R

). (84)

Diagrammatically, the equality can be recast,

with a slight abuse of notation, as in Eq. 83, as

follows

N

+∗

R

V

R

N

+∗

R

¯

R

e

=

N

+∗

R

V

R

N

+∗

R

e

.

Notice that, by the no-restriction hypothesis, for

an admissible V the map V

R

is linear and admis-

sible, and is thus a transformation in [[A

(G)

N

+∗

R

→

A

(G)

R

]]. Moreover,

V

†

a

R

= (V

†

R

a)

N

+∗

R

. (85)

Finally, notice that by deﬁnition V

†

R

e

R

= e

N

+∗

R

,

namely V

R

∈ [[A

(G)

N

+∗

R

→ A

(G)

R

]]

1

. The argument

can be repeated for the case where N

+∗

g

is not

ﬁnite, as long as N

+∗

g

⊂ G.

We remark that in the case of Quantum or

Fermionic cellular automata, the condition for

no causal inﬂuence is equivalent to no-signalling,

while in the case of Classical cellular automata

no causal inﬂuence is strictly stronger than no-

signalling. In a diﬀerent context, a diﬀerence be-

tween the two notions was pointed out in Ref. [55]

5.4 Block decomposition

Let us now consider the GUR (G, A, W

†

) where

we set G = H × {0, 1}, and W = V

H

0

⊗ V

−1

H

1

,

for an arbitrary GUR (H, A, V

†

), H

i

denoting a

shorthand for (H, i). We observe that

W = V

H

0

⊗ V

−1

H

1

= (V

H

0

⊗ I

H

1

)S

H

(V

−1

H

0

⊗ I

H

1

)S

H

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 38

Then, by Eq. (76), given R ⊆ A

G

and a ∈

[[

¯

A

(G)

R

¯

C]]

QR

, with R ∩ G = (R

0

, 0) ∪ (R

1

, 1),

W

†

a

RC

= (W

†

a)

R

0

C

=[S

†

H

(V

−1

H

0

⊗ I

H

1

)

†

S

†

H

(V

H

0

⊗ I

H

1

)

†

]a

RC

=[S

†

H

(V

−1

H

0

⊗ I

H

1

)

†

S

†

H

]a

0

R

00

C

=[S

†

H

(V

−1

H

0

⊗ I

H

1

)

†

S

†

N

+∗

R

0

∪R

1

]a

0

R

00

C

=[S

†

H

(V

−1

H

0

⊗ I

H

1

)

†

S

†

N

+∗

R

0

∪R

1

(V

H

0

⊗ I

H

1

)

†

]a

RC

,

where a

0

:

= (V

H

0

⊗ I

H

1

)

†

a, R

0

:

=

(N

+∗

R

0

, 0) ∪ (N

+

R

1

, 1), and R

00

:

= (N

+∗

R

0

, 0)∪(R

1

, 1).

Now, deﬁning for S ∈ R

(H)

S

0

S

:

=

Y

h∈S

S

0

h

, (86)

S

0

h

:

= (V

H

0

⊗ I

H

1

)S

h

(V

−1

H

0

⊗ I

H

1

), (87)

thanks to eq. (77) we have

(V

−1

H

0

⊗ I

H

1

)

†

S

†

N

+∗

R

0

∪R

1

(V

H

0

⊗ I

H

1

)

†

= S

0†

N

+∗

R

0

∪R

1

.

We remark that, since (V

H

0

⊗I

H

1

)·(V

−1

H

0

⊗I

H

1

)

is an automorphism of [[A

G

→ A

G

]]

QR

, one has

S

0

f

S

0

g

= S

0

g

S

0

f

, ∀f, g ∈ H. (88)

Thus, the product

Q

h∈N

+∗

R

S

0

h

†

in Eq. (86) is well

deﬁned. Notice also that, by deﬁnition

S

0

h

S

0

h

= I

G

. (89)

Since S

0

h

∈ [[A

(G)

(N

+

h

,0)∪(h,1)

→ A

(G)

(N

+

h

,0)∪(h,1)

]]

Q1

(see Fig. 1), one has that S

0

N

+∗

R

0

∪R

1

∈ [[A

(G)

S

→

A

(G)

S

]]

Q1

, with

S

:

= (N

+

N

+∗

R

0

∪ N

+

R

1

, 0) ∪ (N

+∗

R

0

∪ R

1

, 1).

Then we can write

W

†

a

RC

= S

†

H

S

0†

N

+∗

R

0

∪R

1

a

RC

= S

†

N

+

N

+∗

R

0

∪N

+

R

1

∪N

+∗

R

0

∪R

1

S

0†

N

+∗

R

0

∪R

1

a

RC

.

However, due to the structure of W

†

, one actually

has

(S

0

N

+∗

R

0

∪R

1

⊗ I

C

)

†

a

RC

= a

00

(N

+

R

1

,0)∪(N

+∗

R

0

,1)C

,

g

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g

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gg

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=

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Figure 1: In the top picture we provide a graphical rep-

resentation of the transformation S

0

g

, with input wires

on the top left and output on the bottom right. In the

bottom picture we provide a graphical illustration of the

identity S

0

g

S

0

g

0

= S

0

g

0

S

0

g

.

and ﬁnally this implies that

W

†

a

RC

= (S

†

N

+∗

R

0

∪N

+

R

1

S

0†

N

+∗

R

0

∪R

1

⊗ I

†

C

)a

RC

.

(90)

We now observe that

W

−1

= V

−1

H

0

⊗ V

H

1

= S

H

W S

H

,

and thus, by Eq. (90), it is

W

−1†

a

RC

= (S

0

N

+∗

R

1

∪R

0

⊗ I

C

)

†

a

0

R

0

C

= (S

0†

N

+∗

R

1

∪R

0

S

†

R

0

∪R

1

⊗ I

†

C

)a

RC

, (91)

where a

0

R

0

C

= [S

†

a]

(R

1

,0)∪(R

0

,1)C

. Let us then

consider R

1

= ∅. In this case for [a

(R,0)C

] ∈

[[

¯

A

(G)

(R,0)

¯

C]], being R = (R, 0) ∪ (∅, 1), it holds that

[(V

†

a)

(N

+∗

R

,0)C

] = S

†

N

+∗

R

S

0†

N

+∗

R

⊗ I

†

C

[a

(R,0)C

],

(92)

[(V

−1

†

a)

(N

+

R

,0)C

] = S

0†

R

S

†

R

⊗ I

†

C

[a

(R,0)C

]. (93)

In view of the above results, we now consider the

maps V A V

−1

for A ∈ [[A

(R,0)

C → A

(R,0)

C]]

R

,

with R ∈ R

(H)

. First of all, we observe that

(V A V

−1

) ⊗ I

H

1

= W (A ⊗ I

H

1

)W

−1

is a local transformation in ∈ [[A

(G)

(N

+

R

,0)

C →

A

(G)

(N

+

R

,0)

C]]

QR

. Let then a ∈ [[

¯

A

(G)

˜

S

¯

C]]

R

for S ∈

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 39

R

(H)

. We have the following (see Appendix C)

[(V A V

−1

) ⊗ I

H

1

]

†

a

˜

SC

=[W (A ⊗ I

H

1

)W

−1

]

†

a

˜

SC

=(S

0†

R

S

†

R

⊗ I

†

C

)(A ⊗ I

H

0

)

†

(S

†

R

S

0†

R

⊗ I

†

C

)a

˜

SC

.

Finally, since this holds for every a, we have that

for A ∈ [[A

(R,0)

C → A

(R,0)

C]]

R

(V A V

−1

) ⊗ I

H

1

=(S

0

R

⊗ I

C

)(I

H

0

⊗ A )(S

0

R

⊗ I

C

). (94)

The above relation, considering Eq. (86), shows

that

N

+

R

⊆

[

g∈R

N

+

g

.

Along with Eq. (81) the latter proves that actu-

ally

N

+

R

=

[

g∈R

N

+

g

. (95)

Analogously, since N

−

R

= {g

0

| ∃g ∈ R, g ∈ N

+

g

0

},

it is

N

−

R

= {g

0

| ∃g ∈ R, g

0

∈ N

−

g

} =

[

g∈R

N

−

g

. (96)

Let us ﬁnally consider the maps (V

−1

BV ) for

B ∈ [[A

(R,1)

C → A

(R,1)

C]]

R

, with R ∈ R

(H)

.

In this case, using Eq. (89) and manipulating

Eq. (94) we obtain

I

H

0

⊗ (V

−1

BV )

= (S

0

N

+∗

R

⊗ I

†

C

)(B ⊗ I

H

1

)(S

0

N

+∗

R

⊗ I

C

).

(97)

Since the S

0

h

’s that do not commute with B ⊗

I

H

1

in Eq. (97) correspond to h ∈ N

−

g

for some

g ∈ R, namely h ∈ N

−

R

, we can conclude that

N

+∗

R

⊆ N

−

R

. (98)

This observation is suﬃcient to prove that N

±∗

R

=

N

∓

R

, as anticipated in the previous section.

Lemma 52. Let (G, A, V

†

) be a GUR. Then

N

±∗

R

= N

∓

R

. (99)

Proof. We remind that g ∈ N

±

f

iﬀ f ∈ N

∓

g

. Now,

considering Eq. (98) for R = g we have that

g

0

∈ N

−∗

g

⇔ g ∈ N

+∗

g

0

⇒ g ∈ N

−

g

0

⇔ g

0

∈ N

+

g

.

This implies that N

−∗

g

⊆ N

+

g

. Exchanging V

and V

−1

, we can conclude that N

+∗

g

⊆ N

−

g

⊆

N

+∗

g

, i.e. N

+∗

g

= N

−

g

. The thesis follows invoking

Eqs. (95) and (96), which then give us

N

−

R

= N

+∗

R

.

In the remainder we will then use N

±

R

and P

±

R

,

and abandon the symbols N

±∗

R

and P

±∗

R

.

We observe that the block decomposition of

V ⊗ V

−1

that we achieved through the block

transformations S

0

g

is a generalisation of a result

that was proved for quantum cellular automata

in Ref. [32]. Local rules of other kinds are used

in the literature, with nice properties that the

present block decomposition lacks, such as com-

posability (see e.g. Ref [56]). However, for later

purposes the block form of Eq. (86) is particularly

suitable in our case, to cope with update rules in

theories without local discriminability.

6 Homogeneity

The principle of homogeneity, as it is usually for-

mulated in physics, regards equivalence of points

in space-time. Here, however, space-time is not

the background for occurrence of physical events,

on the contrary it emerges as a convenient de-

scription of relations between information pro-

cessing events. We then provide a deﬁnition

of homogeneity for a global update rule, that

regards the way in which diﬀerent systems are

treated by the evolution. In particular, roughly

speaking, every system must be treated in the

same way by the GUR.

It would be easy to state the homogeneity prin-

ciple if we knew in advance that the set G has

some geometric structure that is invariant under

the action of a group—technically speaking, if G

were a symmetric space—so that we could de-

ﬁne pairs of homologous regions as those mapped

into one another by some group element. In this

case, we could provide a criterion for the evolu-

tion to treat every system in the same way, con-

sisting in the existence of some representation of

the group of symmetries of the set G that con-

nects the evolution of every two homologous re-

gions. The group in this case would provide the

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 40

two following ingredients that are needed for the

deﬁnition of homogeneity: i) a notion of “running

the same test on diﬀerent regions”, and ii) a cri-

terion for equivalence of two homologous regions

provided by the requirement that the events of

the same test in the two regions have equal prob-

abilities, if the regions are prepared in the same

state.

However, we are in a situation where the struc-

ture of G must emerge from the evolution given

by the global rule V . The detailed idea is thus to

turn the above argument on its head, and deﬁne

two regions to be homologous if they are “treated

in the same way" by V . The precise construction

is the following.

First of all, in order for the dynamics to treat

two regions in the same way, the two regions need

to have the same structure, in such a way that we

can make sense of “performing the same range of

tests in the two diﬀerent regions”. This concept is

captured by the extension of the notion of oper-

ational equivalence (see deﬁnition 2) to regions.

Intuitively, two operationally equivalent regions

must correspond to operationally equivalent sys-

tems. However, they also need to“have the same

structure”, where the term structure refers to the

fact that a system A

R

associated with a region

R is decomposed in a preferred way into subsys-

tems A

g

i

corresponding to the cells that compose

the region R = {g

1

, g

2

, . . . , g

k

}. The notion of

operational equivalence of regions must then also

capture the idea that two equivalent regions ad-

mit equivalent decompositions.

The deﬁnition is thus recursive, starting from

elementary regions R = {g}, and then deﬁning

equivalence of R

1

and R

2

as the equivalence of

any strict subregion S

1

⊂ R

1

with a strict subre-

gion S

2

⊂ R

2

, and viceversa.

We recall Remark 5, namely that for the re-

mainder every elementary region g ∈ G corre-

sponds to some non-trivial system, A

g

6' I.

Deﬁnition 41. Two elementary regions g and h

are operationally equivalent if A

g

∼

=

A

h

. Two ﬁ-

nite regions R

1

, R

2

⊆ G are operationally equiv-

alent if A

R

1

' A

R

2

, and for every S

1

⊂ R

1

there

exists S

2

⊂ R

2

, and viceversa for every S

2

⊂ R

2

there exists S

1

⊂ R

1

, such that A

S

1

and A

S

2

are

operationally equivalent.

Two operationally equivalent regions have the

same structure, namely they can be decomposed

into pairwise equivalent subsystems, as we now

prove.

Lemma 53. An elementary region R

1

= {g}

cannot be operationally equivalent to a non-

elementary one R

2

= {h

1

, . . . , h

k

}.

Proof. The statement is trivial, since R

1

= {g

1

}

does not have proper subregions, while any non-

elementary region R

2

does, so operational equiv-

alence cannot hold.

Lemma 54. The regions R

1

and R

2

are op-

erationally equivalent if and only if R

1

=

{g

1

, g

2

, . . . , g

k

}, R

2

= {h

1

, h

2

, . . . h

k

}, and A

g

i

∼

=

A

h

i

.

Proof. The condition is clearly suﬃcient, as one

can immediately realise by considering the re-

versible transformation T

S

∈ [[A

S

1

→ A

S

2

]]

1

de-

ﬁned as T

S

:

=

N

g

j

∈S

1

T

j

, with T

j

∈ [[A

g

j

→

A

h

j

]]

1

reversible for every j = 1, . . . , k. We now

prove that the condition is necessary. Let R

1

and R

2

be operationally equivalent. By deﬁni-

tion, and by lemma 53, for every g

l

∈ R

1

there

is h

m

∈ R

2

such that A

g

l

∼

=

A

h

m

, and viceversa.

We now introduce the equivalence relation

g

i

∼ g

j

⇔ A

g

i

∼

=

A

g

j

.

We can then partition R

1

and R

2

into equivalence

classes [g

i

] and [h

j

], respectively, obtaining R

1

=

S

g

i

l

[g

i

l

] and R

2

=

S

h

i

l

[h

i

l

]. Let us start consid-

ering the case where R

1

contains a unique equiv-

alence class: R

1

= [g

1

]. Then R

2

must contain at

least one element h

1

with A

h

1

∼

=

A

g

1

. Moreover,

R

2

cannot contain h

0

such that h

0

6∼ h

1

, because

h

0

must be equivalent to an elementary subregion

of R

1

, by lemma 53. Then, R

2

= [h

1

]. Finally,

since A

⊗h

g

1

∼

=

A

R

2

∼

=

A

R

1

∼

=

A

⊗k

g

1

, by lemma 2

it must be h = k. In the general case where

R

1

contains more than one equivalence class, one

can simply consider the subregions S

1

l

:

= [g

i

l

].

Clearly, for each S

1

l

there must be S

2

l

= [h

j

l

],

with |S

2

l

| = |S

1

l

|.

Remark 7. Notice that, as a consequence of the

above result, two operationally equivalent regions

R

1

, R

2

∈ R

(G)

have the same cardinality |R

1

| =

|R

2

|.

Given two operationally equivalent regions

R

1

, R

2

⊆ G, we now have a way of deﬁning the

notion of “running the same test”, that resorts to

the identiﬁcation of a canonical isomorphism of

the two regions.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 41

Deﬁnition 42. Given two operationally equiv-

alent regions R

1

= {g

1

, g

2

, . . . , g

k

}, R

2

=

{h

1

, h

2

, . . . , h

k

} ∈ R

(G)

, let us choose a special re-

versible transformation U ∈ [[A

R

1

→ A

R

2

]] such

that U =

N

k

i=1

U

i

, where U

i

∈ [[A

g

i

→ A

h

i

]]

is reversible for every i. We then say that R

1

is operationally equivalent to R

2

through U , and

that {B

(2)

l

}

n

l=1

⊆ [[A

R

2

C → A

R

2

C]] represents the

same test as {B

(1)

l

}

n

l=1

⊆ [[A

R

1

C → A

R

1

C]] if for

every 1 ≤ l ≤ n one has

B

(2)

l

= (U ⊗ I

C

)B

(1)

l

(U

−1

⊗ I

C

). (100)

In order to discriminate the way in which two

systems A

g

1

and A

g

2

evolve, one needs to per-

form a testing-scheme, consisting of a stimulus-

test {A

(x)

k

}

m

k=1

⊆ [[A

R

x

C → A

R

x

C]]

Q

on the re-

gion R

x

3 g

x

, possibly involving an ancillary sys-

tem C, and, after the evolution step V , a control-

test {B

(x)

l

}

n

l=1

⊆ [[A

N

+

R

x

C → A

N

+

R

x

C]]

Q

on the re-

gion N

+

R

x

(where x ∈ {1, 2}) and the ancilla C, in

order to detect a “response”.

Two operationally equivalent regions are ho-

mologous if there is a GUR T that interchanges

them while preserving equivalence of local tests

of ﬁnite regions. In rigorous words, we have the

following deﬁnition.

Deﬁnition 43 (Homologous regions). Let

(G, A, V

†

) be a GUR, and let R

1

, R

2

∈ R

(G)

be operationally equivalent regions through T

i

∈

[[A

R

1

→ A

R

2

]], such that N

+

R

1

and N

+

R

2

are op-

erationally equivalent through T

o

∈ [[A

R

0

1

→

A

R

0

2

]]. We say that R

2

is homologous to R

1

if

there exists an admissible GUR (G, A, T

†

), such

that for all schemes ({A

(i)

k

}

m

k=1

, {B

(i)

l

}

n

l=1

) ⊆

[[A

R

i

C → A

R

i

C]]

Q

× [[A

N

+

R

i

C → A

N

+

R

i

C]]

Q

rep-

resenting the same pair of tests for i = 1, 2, one

has B

(1)

l

V A

(1)

k

= T

−1

B

(2)

l

V A

(2)

k

T . We de-

note this relation as R

2

 R

1

.

We remark that, while the relation  deﬁned

above clearly depends on V , we will generally not

make this fact explicit in the notation, as we will

always consider contexts where a given V is con-

sidered, and there will not be room for ambiguity.

On the other hand, when we want to specify the

GUR T that makes R

1

homologous to R

2

, we

write R

2



T

R

1

.

Lemma 55. If R

2



T

R

1

then R

1



T

−1

R

2

.

Proof. For every scheme ({A

(i)

k

}

m

k=1

, {B

(i)

l

}

n

l=1

)

with A

(1)

k

= (T

−1

i

⊗ I

C

)A

(2)

k

(T

i

⊗ I

C

) and

B

(1)

l

= (T

−1

o

⊗ I

C

)B

(2)

l

(T

o

⊗ I

C

), one has

B

(1)

l

V A

(1)

k

= T

−1

B

(2)

l

V A

(2)

k

T . Now, invert-

ing T on the right and T

−1

on the left, one

obtains B

(2)

l

V A

(2)

k

= T B

(1)

l

V A

(1)

k

T

−1

.

Lemma 56. Let V be a global rule, and let the

region R

2

be homologous to R

1

through T . Then

T

−1

V T = V .

Proof. It is suﬃcient to consider the special case

where the test scheme is ({A

(1)

0

}, {B

(1)

0

}), with

A

(1)

0

= I

R

1

C

and B

(1)

0

= I

N

+

R

1

C

. Then, we

have A

(2)

0

= I

R

2

C

, B

(2)

0

= I

N

+

R

2

C

, and V =

T

−1

V T .

Lemma 57. Let V be a global rule, and let R

2



R

1

for some suitable T . Then

T [[A

(G)

R

1

C → A

(G)

R

1

C]]T

−1

= [[A

(G)

R

2

C → A

(G)

R

2

C]],

T [[A

(G)

N

+

R

1

C → A

(G)

N

+

R

1

C]]T

−1

= [[A

(G)

N

+

R

2

C → A

(G)

N

+

R

2

C]].

Proof. Let us choose the test scheme as

({A

(1)

k

}

m

k=1

, {B

(1)

0

}), with B

(1)

0

= I

R

0

1

C

. Then,

we have B

(2)

0

= I

R

0

2

C

, and

V A

(1)

k

= T

−1

V A

(2)

k

T

=T

−1

V (T

i

⊗ I

C

)A

(1)

k

(T

−1

i

⊗ I

C

)T

=T

−1

V T T

−1

(T

i

⊗ I

C

)A

(1)

k

(T

−1

i

⊗ I

C

)T

=V T

−1

(T

i

⊗ I

C

)A

(1)

k

(T

−1

i

⊗ I

C

)T ,

where in the last equality we used the result of

lemma 56. If we now invert V to the left on both

sides, we obtain

A

(1)

k

= T

−1

(T

i

⊗ I

C

)A

(1)

k

(T

−1

i

⊗ I

C

)T ,

and ﬁnally

T A

(1)

k

T

−1

=(T

i

⊗ I

C

)A

(1)

k

(T

−1

i

⊗ I

C

) = A

(2)

k

.

A similar argument leads us to conclude that

T B

(1)

k

T

−1

=(T

o

⊗ I

C

)B

(1)

k

(T

−1

o

⊗ I

C

) = B

(2)

k

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 42

We remark that, straightforwardly,

(R

2

 R

1

) ⇒ (A

R

1

∼

=

A

R

2

). (101)

Now, the discrimination of systems g

1

and g

2

is successful if there exists a region R

1

3 g

1

such

that for any operationally equivalent region R

2

3

g

2

, the region R

2

is not homologous to R

1

. In

precise terms, we have the following deﬁnition.

Deﬁnition 44 (Absolute discrimination). The

evolution given by the global rule V discriminates

two systems g

1

, g

2

∈ G if for any GUR T there

exists a region R

1

, containing g

1

, for which there

is no region R

2

, containing g

2

, that is homologous

to R

1

through T , with g

1



T

g

2

.

In simple words, the evolution given by the

global rule V allows for discrimination of the two

systems g

1

, g

2

∈ G if there exists a test that gives

diﬀerent probabilities depending on whether it is

applied at g

1

or at g

2

, or if the tests that one can

run on R

1

cannot be performed on any region R

2

containing g

2

, or viceversa.

Now, the notion of discrimination of systems g

1

and g

2

with respect to a third system e is given

as follows.

Deﬁnition 45 (Relative discrimination). The

evolution V discriminates two systems g

1

, g

2

∈ G

relatively to a reference system e, with e ∈ G, if

for every T there exists a region R

1

, containing

{g

1

, e}, for which there is no region R

2

, contain-

ing {g

2

, e}, homologous to R

1

through T with

e 

T

e and g

1



T

g

2

.

We can now require the evolution described

by a global rule to be homogeneous by a precise

statement.

Principle 1 (Homogeneity). The global rule V

discriminates any two systems g

1

6= g

2

∈ G, rela-

tively to an arbitrary reference system e ∈ G, but

not absolutely.

Before providing a convenient restatement of

the principle, we analyse some of its aspects and

consequences. In the ﬁrst place, the principle

says that if we do not choose a reference element

e ∈ G, every two systems g

1

and g

2

cannot be

discriminated. If we take the negation of deﬁni-

tion 44, we obtain a straightforward consequence

of homogeneity.

Lemma 58. Let V satisfy the homogeneity prin-

ciple 1. For every pair g

1

, g

2

∈ G, there exists

T such that for every choice of region R

1

3

g

1

, there exists an operationally equivalent re-

gion R

2

3 g

2

through T

i

∈ [[A

R

1

→ A

R

2

]] such

that also N

+

R

1

and N

+

R

2

are operationally equiv-

alent through T

o

∈ [[A

R

0

1

→ A

R

0

2

]], and for all

schemes ({A

(i)

k

}

m

k=1

, {B

(i)

l

}

n

l=1

) representing the

same tests for i = 1, 2, one has B

(1)

l

V A

(1)

k

=

T B

(2)

l

V A

(2)

k

T

−1

.

Proof. The statement follows negating deﬁni-

tion 44.

We can now draw a ﬁrst non trivial conse-

quence from the statement of principle 1.

Lemma 59. For a homogeneous GUR

(G, A, V

†

), all local systems A

g

for g ∈ G

are operationally equivalent.

Proof. Let us consider two elements g

1

, g

2

∈ G.

As a consequence of the homogeneity principle 1,

for the special choice R

1

= {g

1

}, there exists

R

2

3 g

2

such that R

2

 {g

1

} for a suitable T .

Now, by lemma 54, it must be |R

2

| = |{g

1

}| = 1,

i.e. R

2

= {g

2

}. Thus, we have

A

g

2

∼

=

A

g

1

.

Corollary 15. If the region R

2

containing g

2

is

homologous to {g

1

} through T , then R

2

= {g

2

},

and

T

−1

[[A

(G)

g

1

C → A

(G)

g

1

C]]T = [[A

(G)

g

2

C → A

(G)

g

2

C]].

(102)

Proof. By the proof of lemma 59 one has that

R

2

= g

2

. The thesis then follows from lemma 57.

The second result that we prove is a technical

lemma that will play a crucial role in the following

results.

Lemma 60. Let (G, A, V

†

) satisfy the homo-

geneity requirement. Then for every region R one

has A

R

=

N

x∈R

A

g

x

∼

=

A

⊗|R|

0

, where A

0

∼

=

A

g

for

every g ∈ G.

Proof. By lemma 59 we know that for every g ∈

G one has A

g

∼

=

A

0

. The remaining part of the

statement thus follows straightforwardly.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 43

Lemma 61. Let V be non trivial and satisfy the

homogeneity requirement, and R

2

 R

1

. Then

|R

1

| = |R

2

|.

Proof. This follows from lemmas 54 and 60, since

R

1

and R

2

are operationally equivalent by deﬁ-

nition.

Let now g

1

and g

2

be an arbitrary pair of sys-

tems in G, and V a homogeneous GUR. By the

homogeneity principle 1, there exists T such that

for every region R

1

3 g

1

there is a region R

2

3 g

2

homologous through T to R

1

. The set of these

transformations is denoted as T

V

, i.e.

T

V

:

= {T | ∃g

1

, g

2

∈ G

: ∀R

1

3 g

1

∃R

2

3 g

2

, R

2



T

R

1

}.

Lemma 62. Let V satisfy homogeneity, and let

T ∈ T

V

. Then there exists a permutation π :

G → G such that if R

2



T

R

1

it is

R

2

= π(R

1

) (103)

Proof. By deﬁnition, if T ∈ T

V

there exist

g

1

, g

2

∈ G such that for every S

1

3 g

1

there

is S

2

3 g

2

such that S

2



T

S

1

. Let now

R

2



T

R

1

, and f

1

∈ R

1

. Consider the ﬁnite

region V

1

:

= {f

1

, g

1

}. Then there is a region

V

2

3 g

2

such that V

2



T

V

1

, and by lemma 61,

one has |V

2

| = |V

1

| = 2. Consequently, it must

be V

2

= {f

2

, g

2

}. Moreover, since we know that

g

2



T

g

1

, by deﬁnitions 41, 42, and 43, it is

f

2



T

f

1

. Thus, for every f

1

∈ R

1

there is a

unique f

2

∈ R

2

such that f

2



T

f

1

. We then

deﬁne π : G → G by setting π(f

1

)

:

= f

2

. By

lemma 55 the map π is both injective and surjec-

tive, and thus it is a permutation of G.

In the following, given T ∈ T

V

, we will de-

note T = T

π

where π is the permutation of G in

lemma 62.

Lemma 63. Let (G, A, V

†

) be a homogeneous

GUR, and π ∈ Π

V

. Then the GUR T

π

is unique.

Proof. Let T

π

, S

π

∈ T

V

correspond to the same

permutation π. Let a ∈ [[

¯

A

G

¯

C]]

QR

. Then by

lemma 28 one has a = A

†

e

G

0

, with A ∈ [[A

G

C →

A

G

C]]

QR

. Thus

(S

π

T

−1

π

)

†

a = [(S

π

T

−1

π

)

†

⊗ I

†

C

]A

†

e

G

0

= (T

−1†

π

S

†

π

⊗ I

†

C

)A

†

(S

−1†

π

T

†

π

⊗ I

†

C

)e

G

0

,

where G

0

= G ∪ h

0

and A

h

0

∼

=

C. Now, by deﬁ-

nition

(S

†

π

⊗ I

†

C

)A

†

(S

−1†

π

⊗ I

†

C

)

= (T

†

π

⊗ I

†

C

)A

†

(T

−1†

π

⊗ I

†

C

)

= (T

†

i

⊗ I

†

C

)A

†

(T

−1†

i

⊗ I

†

C

),

which implies

(S

π

T

−1

π

)

†

a = a.

Since the latter holds for every C and every a, we

conclude that S

π

= T

π

.

Corollary 16. The set T

V

= {T

π

} is a repre-

sentation of a group Π

V

= {π | T

π

∈ T

V

} of

permutations of G.

We can summarise the above results as follows.

For every pair g

1

, g

2

there exists a reversible map

π : G → G such that π(g

1

) = g

2

and a reversible

transformation T

π

of [[A

G

]]

Q

such that T

−1

π

·T

π

is

an automorphism of [[A

G

→ A

G

]]

Q

and for every

R ⊂ G and every system C

[[A

(G)

R

C →A

(G)

R

C]]

Q∗

= T

−1

π

[[A

(G)

π(R)

C → A

(G)

π(R)

C]]

Q∗

T

π

,

and T

π

leaves the GUR V invariant, i.e.

T

−1

π

V T

π

= V .

The transformations π clearly form a group Π

V

whose action Π

V

× G → G is transitive.

We now proceed considering the second part of

the homogeneity principle, i.e. the statement that

discrimination of every pair g

1

6= g

2

is possible

relative to some third element e ∈ G. The main

consequence is that the action Π

V

× G → G is

free.

Lemma 64. Let V satisfy the homogeneity prin-

ciple 1. If e 6= π ∈ Π

V

, there is no element g of

G such that π(g) = g.

Proof. Let T

π

∈ T

V

. Suppose that there is

g ∈ G such that π(g) = g. Then for every re-

gion R

1

3 g, the region R

2

:

= π(R

1

) contains g

and one has R

2



T

R

1

, with g 

T

g. This is

in contradiction with deﬁnition 45, i.e. with the

homogeneity principle 1.

Corollary 17. Let Π

V

be the group of permu-

tations of G such that T

V

is a representation of

Π

V

. The action of Π

V

on G is regular.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 44

We can now summarise all the results that

we drew from homogeneity as follows: for every

g

1

, g

2

∈ G there exists a permutation π of G such

that π(g

1

) = g

2

, and a GUR T

π

of [[

¯

A

G

]]

QR

such

that for every R ∈ R

(G

0

)

and C

[[A

(G)

R

C → A

(G)

R

C]]

Q∗

= T

−1

π

[[A

(G)

π(R)

C → A

(G)

π(R)

C]]

Q∗

T

π

, (104)

and T

π

leaves the evolution invariant, i.e.

T

−1

π

V T

π

= T .

The transformations π form a group Π

V

that acts

regularly (i.e. transitively and freely) on G.

Now, it is clear that, chosen e ∈ G, the ele-

ments of G can be identiﬁed with the elements

of Π

V

, as the action of Π

V

on G is regular.

In other therms, G is a principal homogeneous

space for Π

V

, and there is a bijection Π

V

↔ G.

However, the homogeneity principle implies an

even stronger result: the set G can be seen as

the set of vertices of a special Cayley graph of

Π

V

. The following arguments, largely inspired to

Refs. [25, 46], make the mentioned result rigor-

ous.

As a consequence of homogeneity, one has

π(g)  π(g

0

) ⇔ g  g

0

. (105)

In other words, N

±

π(g)

= π(N

±

g

) for π ∈ Π

V

, as

can be checked starting from the deﬁnition of N

±

g

.

The proof is provided in Appendix D.

Let now g ∈ G and π ∈ Π

V

such that π(e) = g.

One can then label the elements g

0

∈ N

±

g

by

π

−1

(g

0

) ∈ N

±

e

= {h

±1

i

}

k

i=1

. Notice that for the

moment we have no reason to claim that k < ∞.

We will associate all the pairs (g, g

0

) ∈ E with

the color h

i

if g = π(e) and g

0

= π(h

i

) for

some π ∈ Π

V

2

. This construction enriches the

graph Γ(G, E), which becomes a vertex-transitive

coloured directed graph, with colours correspond-

ing to the labels h

i

∈ N

+

e

Given two elements g, g

0

∈ G, a path p from g

to g

0

is an array of elements p = (g

1

, g

2

, . . . , g

k

) ∈

2

In principle, the colour of an edge (g, g

0

) might

be ambiguously deﬁned if there existed two permuta-

tions π, σ ∈ Π

V

such that π

−1

(g) = σ

−1

(g) = e and

π

−1

(g

0

), σ

−1

(g

0

) ∈ N

±

e

, but π

−1

(g

0

) 6= σ

−1

(g

0

). How-

ever, due to lemma 64, this is never the case, since it

would imply that σπ

−1

(g

0

) 6= g

0

, and thus σπ

−1

6= e, but

[σπ

−1

](g) = σ(e) = g.

G

×k

such that, setting g

0

:

= g and g

k+1

:

= g

0

,

one has

∀0 ≤ i ≤ k (g

i

 g

i+1

) ∨ (g

i+1

 g

i

).

The lenght of a path p, denoted (p) is k +1 if p ∈

G

×k

. In the following we will restrict attention

to the case where Γ(G, E) is connected, i.e. for

every two elements g, g

0

∈ G there exists a path

from g to g

0

. The set of paths from g to g

0

will

be denoted by p

+

g,g

0

. We can now introduce the

graph metric of Γ(G, E) as follows. Let g, g

0

∈ G.

Then we deﬁne

d

Γ

(g, g

0

)

:

= min

p∈p

+

g,g

0

(p)

We remark that d

Γ

(g, g

0

) = d

Γ

[π(g), π(g

0

)] for all

π ∈ Π

V

, since g

1

, . . . , g

k

is a path from g to g

0

if

and only if π(g

1

), . . . , π(g

k

) is a path from π(g)

to π(g

0

) due to condition (105).

We now use the alphabet N

+

e

∪ N

−

e

to form

arbitrary words, obtaining a free group F: com-

position corresponds to word juxtaposition, with

the empty word λ representing the identity, and

the formal rule h

i

h

−1

i

= h

−1

i

h

i

= λ. An element

w = h

p

1

i

1

h

p

2

i

2

. . . h

p

n

i

n

of F —with p

j

∈ {−1, 1}—

thus corresponds to a path on the graph, where

the symbol h

−1

i

∈ N

−

e

denotes a backwards step

along the arrow h

i

. For every h

p

1

i

1

h

p

2

i

2

. . . h

p

m

i

m

=

w ∈ F , one has w

−1

= h

−p

m

i

m

. . . h

−p

2

i

2

h

−p

1

i

1

.

The action M : F × G → G of words w =

h

p

1

i

1

. . . h

p

m−1

i

m−1

h

p

m

i

m

∈ F on elements g ∈ G can be

deﬁned as follows

M(w, g) = M(h

p

1

i

1

h

p

2

i

2

. . . h

p

m

i

m

, g)

:

= M[h

p

2

i

2

. . . h

p

m

i

m

, M(h

p

1

i

1

, g)],

where the action of h

i

∈ S on the elements g ∈ G

is deﬁned as M(h

i

, g) = g

0

if there exists π ∈ Π

V

such that π

−1

(g) = e and π

−1

(g

0

) = h

i

, while

M(h

−1

i

, g) = g

0

if there exists π ∈ Π

V

such that

π

−1

(g) = h

i

and π

−1

(g

0

) = e

3

. In the following

we will use the shortcut

gw

:

= M(w, g).

3

The action is well deﬁned. Suppose indeed that there

are two elements f, f

0

both satisfying the deﬁnition of

M(h

i

, g). This means that there exist π, σ such that

π

−1

(g) = σ

−1

(g) = e, while π

−1

(f) = σ

−1

(f

0

) = h

i

.

However, according to lemma 64, the ﬁrst condition im-

plies that σ = π. By the same lemma, the second condi-

tion then implies that f = f

0

. A similar argument shows

that M (h

−1

i

, g) is well deﬁned.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 45

This notation is particularly handy, since

g(h

p

1

i

1

h

p

2

i

2

. . . h

p

m

i

m

) = (gh

p

1

i

1

)h

p

2

i

2

. . . h

p

m

i

m

.

For every w ∈ F , and for every g ∈ G and

for every permutation π ∈ Π

V

, we now show that

π(gw) = π(g)w. The ﬁrst step consists in proving

the result for w = h

±1

i

. Let g

0

= gh

i

, namely by

deﬁnition σ

−1

(g) = e and σ

−1

(g

0

) = h

i

for some

(unique) σ ∈ π

V

. By Eq. (105) it is π(g)  π(g

0

).

Moreover, if we deﬁne f

:

= π(g) and f

0

:

= π(g

0

)

we have

h

i

= σ

−1

(g

0

) = σ

−1

π

−1

(f

0

),

e = σ

−1

(g) = σ

−1

π

−1

(f),

namely f

0

= fh

i

. This implies

π(gh

i

) = π(g

0

) = π(g)h

i

. (106)

Similarly, if g

0

= gh

−1

i

by deﬁnition σ

−1

(g) = h

i

and σ

−1

(g

0

) = e for some (unique) σ ∈ π

V

. Thus,

setting again f

:

= π(g) and f

0

:

= π(g

0

), we have

h

i

= σ

−1

(g) = σ

−1

π

−1

(f),

e = σ

−1

(g

0

) = σ

−1

π

−1

(f

0

),

namely f

0

= fh

−1

i

, and

π(gh

−1

i

) = π(g

0

) = π(g)h

−1

i

. (107)

Notice that the last result implies that for every

pair f, g ∈ G, given π ∈ Π

V

such that π(f ) = g,

there is a bijection N

±

f

↔ N

±

g

, given by

N

±

π(g)

= π(N

±

g

). (108)

One can now prove that π(fw) = π(f)w by in-

duction on the length l(w) of the word w. Indeed,

we know that it is true for l(w) = 1. Suppose now

that for l(w) = n − 1 one has π(fw) = π(f)w,

and consider w

0

with l(w

0

) = n. Then w

0

= wh

p

i

with l(w) = n − 1 and p = ±1. We have

π(fw

0

) = π(fwh

p

i

) = π[(fw)h

p

i

]

= π(fw)h

p

i

= π(f)wh

p

i

= π(f)w

0

,

where the induction hypothesis is used in the

fourth equality.

Let us now suppose that for some f ∈ G and

some word w ∈ F one has fw = f . Then for

every f

0

∈ G one can take π such that π(f) = f

0

,

thus obtaining

f

0

w = π(f )w = π(f w) = π(f) = f

0

.

Thus, if a path w ∈ F is closed starting from

f ∈ G, then it is closed also starting from any

other g ∈ G.

We can now easily see that the subset R of F

corresponding to words r such that gr = g for

all g ∈ G is a normal subgroup. Indeed, R is a

subgroup because the juxtaposition of two words

s, s

0

∈ R is again a word ss

0

∈ R, and for every

word s ∈ R also s

−1

∈ R. To prove that R is nor-

mal in F we just show that it coincides with its

normal closure, i.e. for every w ∈ F and every r ∈

R, we have wrw

−1

∈ R. Indeed, deﬁning for arbi-

trary g the element g

0

:

= gw, we have g

0

w

−1

= g,

and thus gwrw

−1

= g

0

rw

−1

= g

0

w

−1

= g, namely

wrw

−1

∈ R.

We thus identiﬁed a normal subgroup R con-

taining all the words r corresponding to closed

paths. If one takes the quotient F/R, one obtains

a group whose elements are equivalence classes of

words in F . If we label an arbitrary element of

G by e, it is clear that the elements of G are in

one-to-one correspondence with the vertices of G,

since for every g ∈ G there is one and only one

class in F/R whose elements lead from e to g.

We can then write g = w for every w ∈ F such

that w represents a path leading from e to g. No-

tice that the elements of F/R, i.e. equivalence

classes of words [w]

R

, correspond to elements of

G, i.e. for every g ∈ G there is a unique class

[w]

R

such that g = [w]

R

. The criterion for class

membership of words is very simple: w ∈ g iﬀ w

connects the vertex e ∈ G to the vertex g ∈ G.

As a consequence of the above arguments, we

can now show that for every π ∈ Π

V

and for every

g = [w]

R

∈ G, one has π(g) = [sw]

R

for a ﬁxed

word s ∈ F . Indeed, consider g

0

= [w

0

]

R

. Let

[w

0

]

R

= π(g

0

). Then w

0

= w

0

w

−1

0

w

0

= sw

0

, with

s

:

= w

0

w

−1

0

, and π(g

0

) = [sw

0

]

R

. Let now f ∈ G.

We can always ﬁnd t ∈ F so that f = [w

0

t]

R

,

e.g. by setting t

:

= w

−1

0

z, f = [z]

R

. Then we have

π(f) = π([w

0

t]

R

) = π(g

0

t) = π(g

0

)t = [sw

0

t]

R

=

[sz]

R

, where we use the deﬁnition π(k)r

:

= [xr]

R

with π(k) = [x]

R

. This deﬁnition clearly makes

sense, since when [a]

R

= [b]

R

and [r]

R

= [u]

R

,

also [ar]

R

= [bu]

R

.

In technical terms, the graph Γ(G, E) =

Γ(G, S) is the Cayley graph of the group G =

F/R. Homogeneity thus implies that the set G is

a group G that can be presented as G = hS|Ri,

where S is the set of generators of G and R is the

group of relators. In the following, if h

i

= h

−1

i

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 46

we will draw an undirected edge to represent h

i

.

The presentation can be chosen by arbitrarily di-

viding S into S

+

⊆ S and S

−

:

= S

−1

+

in such a

way that S

+

∪ S

−

= S. The above arbitrariness

is inherent the very notion of group presentation

and corresponding Cayley graph, and is exploited

in the literature, in particular in the deﬁnition of

isotropy [25, 57].

For convenience of the reader we remind the

deﬁnition of Cayley graph. Given a group G and

a set S

+

of generators of the group, the Cayley

graph Γ(G, S

+

) is deﬁned as the colored directed

graph having vertex set G, edge set {(g, gh) | g ∈

G, h ∈ S

+

}, and a color assigned to each gener-

ator h ∈ S

+

. Notice that a Cayley graph is reg-

ular—i.e. each vertex has the same degree—and

vertex-transitive—i.e. all vertices are equivalent,

in the sense that the graph automorphism group

acts transitively upon its vertices. The Cayley

graphs of a group G are in one to one correspon-

dence with its presentations, with Γ(G, S

+

) cor-

responding to the presentation hS

+

|Ri.

7 Locality

If a GUR were to represent a physical law, we

would like that, in order to determine it, it is

suﬃcient to test it in a ﬁnite region and for a ﬁ-

nite amount of time. This possibility is granted

by homogeneity in conjunction with a second

principle—locality—to which the present section

is devoted. The locality requirement can be

phrased in simple words by stipulating that in

order to calculate the eﬀects of a physical law, i)

it is possible to adopt a reductionist procedure,

decomposing an arbitrary system into elementary

parts and then calculating its evolution by evolv-

ing every part separately, considering only pair-

wise interactions between parts, and ii) if the re-

duction is operated, only ﬁnite systems will be

involved in the calculation for every elementary

part.

In the present section we introduce the precise

statement of the locality principle, and analyse its

main consequences. In order to make this notion

independent of homogeneity, we formulate it for

general admissible GURs. In particular, this im-

plies that the mathematical statement of locality

alone will not allow for determination of a local

GUR by observation of a ﬁnite region.

Before giving a formal deﬁnition, we provide a

few heuristic steps leading to the ﬁnal formula-

tion. Let us start considering an admissible GUR

(G, A, V

†

), with uniformly bounded neighbour-

hood, namely such that there exists k < ∞ so

that for every g ∈ G it is |N

±

g

| ≤ k. We remark

that, since |N

±

g

| ≤ k, it is |P

±

g

| ≤ k

2

. This also

implies that the transformations S

0

g

are transfor-

mations of ﬁnitely many systems.

In the remainder, we will often omit the labels

0, 1 on the wires in diagrams, and adopt the con-

vention that the upper half is labelled 0 and the

lower one 1.

Now, reminding Eq. (92), given a ∈

[[

¯

A

(G)

R

C]]

QR

, one has

[(V

†

a)

(N

−

R

,1)C

] = [{(V

†

R

⊗ I

†

C

)a}

(N

−

R

,1)C

]

= [{(S

0†

N

−

R

⊗ I

†

C

)a}

(N

−

R

,1)C

].

Thus, we have

N

−

R

V

R

=

ψ

P

−

R

S

0

N

−

R

P

−

R

\R

e

N

−

R

N

−

R

e

.

(109)

Notice that, by admissibility of the GUR V , and

thanks to the no-restriction hypothesis,

V

R

∈ [[A

N

−

R

→ A

R

]]

1

.

Moreover, the GUR V can be equivalently de-

ﬁned starting from its action on [[

¯

A

G

]]

LR

, by set-

ting V

†

[a

R

]

:

= [(V

†

R

a)

N

−

R

]. Indeed, following

this deﬁnition, one can prove that V is right-

invertible, by the argument discussed in Ap-

pendix E. We remark that the construction of the

inverse provides a new GUR W .

As a consequence, we have that W

†

V

†

=

V

†

W

†

= I

†

G

for some GUR W , and thus W =

V

−1

. We want to stress here that in proving

this result we only use properties (88), (89), (94)

and (97) of S

0

g

, derived in section 5.4, along with

uniform boundedness of the neighbourhoods for

the GUR V (see Appendix E for the details).

Let us now abstract our focus from the situa-

tion where a GUR (G, A, V

†

) is given. Instead,

we will suppose that, given the pair (G, A), a ﬁ-

nite neighbourhood N

+

g

⊆ G is deﬁned for every

g ∈ G, and for f ∈ G we set

N

−

f

:

= {h ∈ G | f ∈ N

+

h

}.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 47

For X ∈ R

(G)

we then deﬁne N

±

X

:

=

S

g∈X

N

±

g

,

and the sets P

±

g

are then deﬁned exactly as for a

GUR. Finally, we suppose that a family of maps

S

0

g

is deﬁned, with the properties (88) and (89),

and an analogue of (94) and (97).

Deﬁnition 46 (Local rule). A k-local rule

(N

+

, A, S

0

) on G is i) a map N

+

: G → R

(G)

that associates g ∈ G with a ﬁnite region N

+

g

∈

R

(G)

such that |N

+

g

|, |N

−

g

| ≤ k for every g ∈ G;

ii) a map S

0

: g 7→ S

0

g

, where S

0

g

are maps in

[[A

(N

+

g

,0)

A

(g,1)

→ A

(N

+

g

,0)

A

(g,1)

]]

1

with the follow-

ing properties for every f, g ∈ G and R, S ∈ R

(G)

:

1. S

0

g

S

0

g

= I

A

N

+

g

A

g

;

2. S

0

g

S

0

f

= S

0

f

S

0

g

;

3. for every C ∈ [[A

(R,0)∪(S,1)

C → A

(R,0)∪(S,1)

C]]

P

−

R

\N

+

S

0

N

−

R

∪S

N

+

N

−

R

∪S

\R

S

0

N

−

R

∪S

P

−

R

\N

+

S

N

+

S R

C

R

N

+

S

C C

N

−

R S S

N

−

R

S\N

−

R

N

−

R

\S S\N

−

R

=

P

−

R

\N

+

S

N

+

S

C

0

N

+

S

C C

N

−

R

N

−

R

S\N

−

R

, (110)

where S

0

X

:

=

Q

g∈X

S

0

g

for every X ⊆ G.

In the following, we will often omit the sub-

script X in S

0

X

in the diagrams, since X corre-

sponds to the label of the lower half wires. More-

over, in order to make the equations lighter, from

now on we will always write or draw transfor-

mations and eﬀects that do not involve explicitly

the external system C, however it will always be

assumed that the conditions hold also locally on

extended systems, unless explicitly stated other-

wise.

Remark 8. Notice that for a transformation C

of the form C

0

⊗I

(R\R

0

,0)∪(S\S

0

,1)

, due to the def-

inition of S

0

N

−

R

∪S

and the property 2 of S

0

g

, one

has that

P

−

R

\N

+

S

N

+

S

C

0

N

+

S

C C

N

−

R

N

−

R

S\N

−

R

=

P

−

R

\N

+

S

N

+

S

\N

+

S

0

N

+

S

0

C

0

N

+

S

0

C C

N

−

R

0

N

−

R

0

N

−

R

\N

−

R

0

S\N

−

R

=

[P

−

R

∪N

+

S

]\[P

−

R

0

∪N

+

S

0

]

P

−

R

0

\N

+

S

0

S

0

N

+

N

−

R

0

∪S

0

\R

0

S

0

P

−

R

0

\N

+

S

0

N

+

S

0

R

0

C

0

R

0

N

+

S

0

C C

N

−

R

0

S

0

S

0

N

−

R

0

S

0

\N

−

R

0

N

−

R

0

\S

0

S

0

\N

−

R

0

[N

−

R

∪S]\[N

−

R

0

∪S

0

]

.

The ﬁrst property that we prove is the follow-

ing.

Lemma 65. Given a local rule (N

+

, A, S

0

) on

G, the following identities hold:

1. For every S ∈ R

(G)

, and every A ∈ [[A

S

C →

A

S

C]],

(N

+

S

,0)

S

0

N

−

S

(N

+

S

,0)

S

0

N

−

S

(N

+

S

,0)

C

A

C

(S,1) (S,1) (S,1) (S,1)

=

(N

+

S

,0)

A

+

(N

+

S

,0)

C C

(S,1)

; (111)

2. for every R ∈ R

(G)

and every B ∈ [[A

R

C →

A

R

C]]

(P

−

R

,0)

S

0

N

−

R

(P

−

R

\R,0)

S

0

N

−

R

(P

−

R

,0)

(R,0)

B

(R,0)

C C

(N

−

R

,1) (N

−

R

,1) (N

−

R

,1)

=

(P

−

R

,0)

C

B

−

C

(N

−

R

,1) (N

−

R

,1)

. (112)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 48

Proof. The equalities follow taking S = ∅ and

R = ∅ in Eq. (110), respectively.

Remark 9. Given a local rule (N

+

, A, S

0

) we

deﬁne the forward and backward evolution of lo-

cal transformations F ∈ [[A

R

C → A

R

C]] for

a ﬁnite region R ∈ R

(G)

through the expres-

sions (111) and (112), as follows

[V

±

L

(F )]

N

±

R

C

:

= [F

±

]

N

±

R

C

. (113)

Now, once we established the notion of a local

rule and its application to local transformations,

we can deﬁne the property of a global rule of be-

ing reducible to a local rule.

Deﬁnition 47 (Reduction to a local rule). We

say that the GUR (G, A, V

†

) is reducible to a

local rule if there exists a k-local rule (N

+

, A, S

0

)

for some k < ∞ such that for every R ∈ R

(G)

and

every F ∈ [[A

R

C → A

R

C]]

(V ⊗ I

C

)F (V

−1

⊗ I

C

) = V

−

L

(F ) ⊗ I

G\N

−

R

,

(V

−1

⊗ I

C

)F (V ⊗ I

C

) = V

+

L

(F ) ⊗ I

G\N

+

R

.

(114)

We now prove the two main results of this sec-

tion, which relate local rules and GURs in a one-

to-one correspondence.

Theorem 9. Every local rule (N

+

, A, S

0

) on G

identiﬁes a GUR (G

0

, A, W

†

), with G

0

= G ×

{0, 1}, and W

−1

= S

G

W S

G

.

Proof. First of all, one can deﬁne the linear maps

W

†

C

on [[

¯

A

G

¯

C]]

LR

as follows. For a

ZC

∈ [[

¯

A

G

¯

C]]

LR

,

with Z = (R, 0) ∪ (S, 1), we set

W

†

[a

ZC

]

:

= [a

−

(N

−

R

,0)∪(N

+

S

,1)C

],

where

˜

N

+

N

−

R

∪S

e

S\N

−

R

e

N

−

R

a

−

C

N

+

S

P

−

R

\N

+

S

e

\

N

−

R

∪S

e

:

=

˜

N

+

N

−

R

∪S

S

\

N

−

R

∪S

e

S\N

−

R

P

−

R

\N

+

S

0

N

+

N

−

R

∪S

\R

e

N

−

R

N

+

S R

a

C

N

+

S

N

−

R S

P

−

R

\N

+

S

S\N

−

R

N

−

R

\S

e

\

N

−

R

∪S

˜

N

+

N

−

R

∪S

e

.

We omitted the subscript for S and S

0

, as the

systems on which the transformations act are

clear from the context. Finally,

˜

N

+

X

is a short-

hand for N

+

X

\X, and

b

X for X \N

+

X

. Writing a =

A

†

e, for A ∈ [[A

(R,0)∪(S,1)

C → A

(R,0)∪(S,1)

C]],

and using Eq. (110), one can easily check that

the eﬀect a

−

is actually in [[

¯

A

(N

−

R

,0)∪(N

+

S

,1)

¯

C]]. We

remark that the map W

†

is well deﬁned: for

a ⊗ e

W

∈ [a

ZC

], with W = (W

0

, 0) ∪ (W

1

, 1),

and W

0

∩ R = W

1

∩ S = ∅, one has

W

†

[(a ⊗ e

W

)

(R∪W

0

,0)∪(S∪W

1

,1)C

] = W

†

[a

ZC

].

The above identity is veriﬁed exploiting

Eq. (110), along with the identity in Re-

mark 8. As to reversibility, for [a

ZC

] with

Z = (R, 0) ∪ (S, 1), one can deﬁne

Z

†

[a

ZC

]

:

= [a

+

(N

+

R

,0)∪(N

−

S

,1)C

],

where

\

N

+

S

∪R

e

P

−

S

\N

+

R

e

N

+

R

a

+

C

N

−

S

R\N

−

S

e

˜

N

+

N

−

S

∪R

e

:

=

\

N

−

R

∪S

S

˜

N

+

N

−

S

∪R

e

P

−

S

\N

+

R

S

0

N

+

N

−

S

∪R

\S

N

−

S

\R

e

N

+

R S R

a

C

N

−

S R S

R\N

−

S

N

−

S

\R

N

+

N

−

S

∪R

\S

e

˜

N

+

N

−

S

∪R

\

N

−

R

∪S

e

,

and, similarly to what was done for W , check

that Z

†

is well deﬁned. One can now verify that

W

†

Z

†

= Z

†

W

†

= I

[[

¯

A

G

]]

LR

. The above observa-

tions show that W

†

and Z

†

= W

−1†

act isomet-

rically on [[

¯

A

G

]]

LR

, since

ka

RC

k

sup

= kZ

†

W

†

a

RC

k

sup

≤ kW

†

a

RC

k

sup

≤ ka

RC

k

sup

,

ka

RC

k

sup

= kW

†

Z

†

a

RC

k

sup

≤ kZ

†

a

RC

k

sup

≤ ka

RC

k

sup

.

Being isometric, both W

†

and W

−1†

preserve

Cauchy sequences and their equivalence classes.

As a consequence, W

†

and W

−1†

can be uniquely

extended to invertible isometries of [[

¯

A

G

0

]]

QR

,

with Z

†

= W

†−1

. Let now ρ ∈ [[A

G

0

C]], and

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 49

a

(R,0)∪(S,1)C

∈ [[

¯

A

G

0

¯

C]]

LR

. Then

(a

(R,0)∪(S,1)C

|

ˆ

W ρ) = (W

†

a

(R,0)∪(S,1)C

|ρ)

=(S

†

N

−

R

∪S∪N

+

N

−

R

∪S

S

0†

N

−

R

∪S

a

(S,0)∪(T,1)C

|ρ)

=(a

(R,0)∪(S,1)C

|

ˆ

S

0

N

−

R

∪S

ˆ

S

N

−

R

∪S∪N

+

N

−

R

∪S

ρ)

=(a|[

ˆ

S

0

N

−

R

∪S

ˆ

S

N

−

R

∪S∪N

+

N

−

R

∪S

ρ]

|(N

−

R

,0)∪(N

+

S

,1)C

),

and clearly this implies that

ˆ

W ρ

|(R,0)∪(S,1)

is a

state for every R, S ∈ R

(G)

. Then

ˆ

W [[A

G

C]] ⊆

[[A

G

C]]. The same argument holds for

ˆ

W

−1

,

thus

ˆ

W [[A

G

C]] = [[A

G

C]]. Let now A

T C

∈

[[A

G

C → A

G

C]]

L

, with T = (R, 0) ∪ (S, 1). One

can evaluate

ˆ

W

−1

A

T C

ˆ

W by dually evaluating

W

†

A

†

T C

W

−1†

. The calculation is a straightfor-

ward application of Eq. (110), and shows that

ˆ

W

−1

A

T C

ˆ

W ∈ [[A

G

C → A

G

C]]

L

. The same argu-

ment holds for

ˆ

W A

T C

ˆ

W

−1

, and thus

ˆ

W

−1

[[A

G

C → A

G

C]]

L

ˆ

W = [[A

G

C → A

G

C]]

L

.

All the above observations prove that W is a UR.

Finally, being constructed from local transforma-

tions, one can easily check that the family {W

C

|

C ∈ Sys} is an admissible UR. This concludes the

proof that (G

0

, A, W

†

) is a GUR. Now, comparing

the equations deﬁning a

±

(N

±

R

,0)∪(N

∓

S

,1)C

, one can

easily conclude that Z = W

−1

= S

G

W S

G

.

Theorem 10. Let (N

+

, A, S

0

) be a local rule on

G. Then the GUR (G

0

, A, W

†

), with G

0

= G ×

{0, 1} in theorem 9, is of the form W = V ⊗V

−1

,

where (G, A, V

†

) is reducible to (N

+

, A, S

0

).

The detailed proof can be found in Appendix F.

The properties used in the proof are (88), (89),

(94), and (97). The ﬁrst two are required as

items 1 and 2 in the deﬁnition 46 of a local rule,

while the remaining two are a consequence of

item 3 as shown by lemma 65.

In the case of a homogeneous update rule, does

the existence of a local rule allow one to determine

the full (G, A, V

†

) by local tests? One would in-

tuitively expect that the answer is positive, since

S

0

g

= S

0

e

are the same for every g ∈ G, by virtue

of homogeneity, and thus, knowing S

0

e

, one could

calculate the action of V through Eq. (112). On

the other hand, after giving the question another

thought, one can realise that calculating the ac-

tion of V through the maps V

±

L

requires another

piece of information besides knowing S

0

e

. Actu-

ally, determining the cellular automaton with lo-

cal tests also requires knowledge of the regions

N

+

R

for every R ∈ R

(G)

—or, in other words,

knowledge of the structure of the graph of in-

ﬂuence relations. This information is indeed nec-

essary in order to know how to correctly calcu-

late the transformation V

±

L

(A

R

) for a given re-

gion R. Indeed, it might happen that knowledge

of ﬁnitely many closed paths from the elements

g ∈ R is not suﬃcient to reconstruct all the new

closed paths that appear as the size of the region

R increases. This means that one needs inﬁnitely

many rules to identify systems in N

±

g

1

∩N

±

g

2

. How-

ever, this is not the case if closed paths in the

graph of inﬂuence relations can be decomposed

into elementary closed paths having a uniformly

bounded size, let us say by a constant length l. In

this case, knowledge of local rules and of the local

structure of the graph up to some ﬁnite distance

l is suﬃcient to calculate the action of V in an

arbitrary ﬁnite region. This is the reason for the

following deﬁnition.

Deﬁnition 48 (Decomposability into bounded

regions). We say that a global update rule

(G, A, V

†

) is decomposable into bounded regions

if the closed paths of the inﬂuence graph of V

can be decomposed into elementary closed paths

of uniformly bounded length.

We are ﬁnally in position to formulate the lo-

cality principle, which can be thought of as the re-

quirement that ﬁnite, local information is needed

in order to reconstruct the update rule.

Principle 2 (Locality). The GUR (G, A, V

†

) is

reducible to a local rule and decomposable into

bounded regions.

8 Cellular Automata

The principles of homogeneity and locality single

out the special class of global rules that we will

call cellular automata.

Deﬁnition 49. Let (G, A, V

†

) be a global up-

date rule obeying the principles of homogeneity

and locality. We say that (G, A, V

†

) is a cellular

automaton.

Notice that the inﬂuence graph of a cellular

automaton is the Cayley graph of a ﬁnitely pre-

sented group, i.e. a group that is ﬁnitely gener-

ated and can be presented through ﬁnitely many

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 50

relators. This fact has the very important con-

sequence that every cellular automaton deﬁnes

a Cayley graph that is quasi-isometric to some

smooth Riemannian manifold of dimension d ≤ 4

(see [47], sec. IV pag. 90).

An important remark is in order. Thanks to

homogeneity, one can easily show that the local

rule S

0

g

for a cellular automaton on the Cayley

graph Γ(G, S

+

) has the property that S

0

g

= S

0

e

G

for every g ∈ G. Thus, cellular automata can be

identiﬁed by their Cayley graph Γ(G, S

+

) and by

a reversible transformation S

0

e

G

∈ [[A

N

+

e

G

A

e

G

→

A

N

+

e

G

A

e

G

]]

1

.

8.1 Results

We now prove two important theorems regarding

cellular automata. The ﬁrst one shows that, in

order to obey the properties of homogeneity and

locality, a cellular automaton cannot be in the

quasi-local algebra.

Theorem 11. Let |G| = ∞. The homogeneous

and local cellular automaton V is an element of

[[A

G

→ A

G

]]

Q1

if and only if it is trivial: V =

I

G

.

Proof. Let V be in the quasi-local algebra, and

V

nR

n

be a sequence of local transformations in

the class of V . Then for ε > 0 there exists n

0

such that for every n ≥ n

0

one has

kV

nR

n

− V k

sup

≤ ε.

Moreover, for π ∈ Π

V

, one has T

π

V T

−1

π

= V .

Then

kV

nR

n

− T

π

V T

−1

π

k

sup

≤ ε.

Since T

π

is a GUR, it is isometric and thus

kT

−1

π

V

nR

n

T

π

− V k

sup

= kV

nR

n

− T

π

V T

−1

π

k

sup

≤ ε. (115)

Now, this implies that

kV

nR

n

− T

−1

π

V

nR

n

T

π

k

sup

≤ 2ε.

It is easy to prove that, if |G| = ∞, for every

R ∈ R

(G)

there must exist π ∈ Π

V

so that R ∩

π(R) = ∅. Suitably choosing π, condition (115)

then takes the form

kW ⊗ I − I ⊗ W k

sup

= kW ⊗ I − W ⊗ W + W ⊗ W − I ⊗ W k

sup

≤ 2kW − I k

sup

≤ ε.

Being ε arbitrary, we must conclude that W =

I , namely V

nR

n

= 1

∅

.

The second result pertains the following types

of theory.

1. Theories with local discriminability [8, 9,

15, 58], or more generally every theory

where the algebra of transformations A ∈

[[A

1

. . . A

n

→ A

1

. . . A

n

]] is generated by lo-

cal transformations, i.e. transformations of

the form

A

i

∈ I

¯

A

i

⊗ [[A

i

→ A

i

]],

where

¯

A

i

is the composite system made of

all A

1

. . . A

n

except from A

i

. This class in-

cludes, among others, classical information

theory and quantum theory.

2. Theories where the algebra of transforma-

tions A ∈ [[A

1

. . . A

n

→ A

1

. . . A

n

]] is gener-

ated by bipartite transformations, i.e. trans-

formations of the form

A

i,j

∈ I

¯

A

i

¯

A

j

⊗ [[A

i

A

j

→ A

i

A

j

]],

where

¯

A

i

¯

A

j

is the composite system made of

all A

1

. . . A

n

except from A

i

and A

j

. This

case includes the Fermionic theory as well as

real quantum theory.

In the above cases, for groups G with suitable

properties that we will immediately specify, the

admissible local rules for a cellular automaton can

be sought considering ﬁnite-dimensional systems.

In the remainder we will then restrict to the fam-

ilies of theories introduced above.

Before proving this result, we need some pre-

liminary deﬁnition and lemma.

Deﬁnition 50 (Quotient group). Let G, H be

two groups, and suppose that there exists a group

homomorphism ϕ : G → H. Then H is a quo-

tient of G. If H is ﬁnite, it is called a ﬁnite

quotient of G.

The reason why we are interested in ﬁnite quo-

tients is that, under suitable assumptions that

will be made rigorous shortly, the same local rule

deﬁnes an automaton on G and an automaton on

any of its ﬁnite quotients H. This result makes

it much easier to provide unitarity conditions in

all those cases where the ﬁnite quotients of the

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 51

group G satisfy the mentioned requirements. We

now explain the hypotheses in detail, and ﬁnally

prove the main theorem.

We start remarking that if the group G has a

ﬁnite presentation

G = ha

1

, . . . , a

n

|r

1

, . . . , r

k

i,

it is straightforward to verify that any quotient

H of G has a ﬁnite presentation of the form

H = hϕ(a

1

), . . . , ϕ(a

n

)|ϕ(r

1

), . . . , ϕ(r

k

), b

1

, . . . , b

j

i,

where the new relations b

k

do not belong to the

group R, conjugate closure of {ϕ(r

1

), . . . , ϕ(r

k

)}.

When H is a quotient of G, referring to the

presentation above, we will denote by R

0

the

conjugate closure of the group generated by

{φ(r

l

)}

k

l=1

∪ {b

m

}

j

m=1

. Moreover, the neighbour-

hoods of f in Γ(H, φ(S

+

)) will be denoted by N

0±

f

for f ∈ H. We now state two lemmas that we use

in the following. The proofs are provided in Ap-

pendix G

Lemma 66. Let N

+

be a neighbourhood system

corresponding to the Cayley graph Γ(G, S

+

) of a

a ﬁnitely generated group G, and let H = φ(G)

be a ﬁnite quotient of G such that for every

h

a

, h

b

, h

c

, h

d

∈ S

+

one has φ(h

a

h

−1

b

) ∈ R

0

, if and

only if h

a

h

−1

b

∈ R and φ(h

a

h

−1

b

h

c

h

−1

d

) ∈ R

0

if

and only if h

a

h

−1

b

h

c

h

−1

d

∈ R.

1. N

0±

φ(g)

= φ(N

±

g

), and |N

0±

φ(g)

| = |N

±

g

|.

2. P

0±

φ(g)

= φ(P

±

g

), and |P

0±

φ(g)

| = |P

±

g

|.

3. Given f

1

, f

2

∈ H, one can choose g

i

∈

φ

−1

(f

i

) such that N

0±

f

1

∩ N

0±

f

2

= φ(N

±

g

1

∩ N

±

g

2

)

and |N

0±

f

1

∩ N

0±

f

2

| = |N

±

g

1

∩ N

±

g

2

|.

4. Given f

1

, f

2

∈ H, one can choose g

i

∈

φ

−1

(f

i

) such that P

0∓

f

2

∩ N

0±

f

1

= φ(P

∓

g

2

∩ N

±

g

1

)

and |P

0∓

f

2

∩ N

0±

f

1

| = |φ(P

∓

g

2

∩ N

±

g

1

)|.

5. If g

2

6∈ N

±

g

1

and φ(g

2

) ∈ N

0±

φ(g

1

)

, it is P

±

g

1

∩

N

∓

g

2

= ∅.

Lemma 67. In addition to the hypothe-

ses of lemma 66, let φ(h

a

h

−1

b

h

c

h

−1

d

h

e

h

−1

f

) ∈

R

0

iﬀ h

a

h

−1

b

h

c

h

−1

d

h

e

h

−1

f

∈ R for every

h

a

, h

b

, h

c

, h

d

, h

e

, h

f

∈ S

+

. Then

1. Given f

1

, f

2

∈ H with f

2

∈ P

0±

f

1

, one has

P

0±

f

1

∩ P

0±

f

2

= φ(P

±

g

1

∩ P

±

g

2

), for some g

i

∈

φ

−1

(f

i

), with g

2

∈ P

±

g

1

, and |P

0±

f

1

∩ P

0±

f

2

| =

|P

±

g

1

∩ P

±

g

2

|.

2. Let g

2

6∈ P

±

g

1

and φ(g

2

) ∈ P

0±

φ(g

1

)

. Then N

±

g

1

∩

N

±

g

2

= P

±

g

1

∩ P

±

g

2

= ∅.

Corollary 18. Under the hypotheses of

lemma 66, the homomorphism φ is invertible on

N

±

g

and P

±

g

for every g ∈ G.

In order to make the hypotheses of lemmas 66

and 67 clearer, in Fig. 2 we show two examples

of ﬁnite quotients of the group Z

2

, the ﬁrst one

satisfying the hypotheses of lemma 66 but not

those of lemma 67, and the second one satisfying

both.

(a)

(b)

(c)

Figure 2: (a) The Cayley graph of Z

2

presented as

ha, b|aba

−1

b

−1

i; (b) the Cayley graph of the ﬁnite

quotient of Z

2

hφ(a), φ(b)|φ(aba

−1

b

−1

), φ(a)

6

, φ(b)

5

i,

satisfying the hypotheses of lemma 66 but not those

of 67; (c) the Cayley graph of the ﬁnite quotient

of Z

2

hφ(a), φ(b)|φ(aba

−1

b

−1

), φ(a)

8

, φ(b)

7

i, satisfying

the hypotheses of both lemmas 66 and 67. Notice that

we implicitly assume that S

+

contains also a

−1

and b

−1

,

but we do not express the presentation accordingly, to

keep the formulas readable. In principle there should

be two extra generators c, d (and φ(c), φ(d) in the ﬁ-

nite quotients), along with the relations ac and bd (and

φ(ac), φ(bd), accordingly).

We can now prove the following results, which

are the core of the ﬁnal theorem at the end of the

section.

Lemma 68. For theories of type 1, under the

hypotheses of lemma 66, given a homogeneous

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 52

local rule (N

+

, A, S

0

) with neighbourhood sys-

tem N

+

such that |N

+

e

G

| = |S

+

|, the local

rule (N

0+

, A,

˜

S

0

), with

˜

S

0

e

H

:

= S

0

e

G

is well-

deﬁned on the Cayley graph Γ(H, φ(S

+

)), with

N

0+

φ(g)

= φ(N

+

g

). Viceversa, given a local

rule (N

0+

, A,

˜

S

0

) whose neighbourhood system

N

0+

corresponds to a Cayley graph Γ(H, φ(S

+

)),

the local rule (N

+

, A, S

0

), with S

0

e

:

=

˜

S

0

e

is

well-deﬁned on the Cayley graph Γ(G, S

+

), with

φ(N

+

g

) = N

0+

φ(g)

Proof. We remark that, by deﬁnition of the maps

S

0

g

(see Deﬁnition 46) a given choice of S

0

e

H

is in principle compatible with every Cayley

graph with |N

0±

e

H

| = |N

±

e

G

|, i.e. the same num-

ber of generators (we remind that S

0

g

= S

0

e

H

∈

[[A

N

+

e

G

A

e

G

→ A

N

+

e

G

A

e

G

]]

1

). By lemma 66, the

cardinality of N

0+

e

H

is the same as that of N

+

e

G

.

Then

˜

S

0

e

H

= S

0

e

G

is well deﬁned on Γ(H, φ(S

+

))

iﬀ it is well deﬁned on Γ(G, S

+

). What remains

to be proved is that the map

˜

S

0

e

H

= S

0

e

G

identi-

ﬁes a local rule on the system of neighbourhoods

N

0+

—namely that Eq. (110) as well as items 1

and 2 in deﬁnition 46 hold—iﬀ it does on the

system of neighbourhoods N

+

. Item 1 is sat-

isﬁed by

˜

S

0

= S

0

by deﬁnition in both cases.

As to item 2, it is trivial to verify that the only

important feature in order to decide whether it

holds in any neighbourhood system N

0+

e

H

is the

cardinality of N

0+

e

H

∩ N

0+

f

(or N

+

e

G

∩ N

+

g

) for every

f ∈ P

0+

e

H

(or g ∈ P

−

e

G

), since this identiﬁes the

systems where both

˜

S

0

e

H

= S

0

e

G

and

˜

S

0

f

= S

0

g

act non-trivially, namely those where the prod-

ucts

˜

S

0

e

H

˜

S

0

f

or S

0

e

G

S

0

g

might not be commuta-

tive. Lemma 66 thus ensures that item 2 is sat-

isﬁed by

˜

S

0

iﬀ it is by S . Let us then focus on

Eq. (110). For theories of type 1, it is suﬃcient

to verify that Eq. (110) holds for |R| = 1 and

|S| = 0 or |R| = 0 and |S| = 1. Both conditions

easily follow if we consider lemma 66.

Lemma 69. For theories of type 2, under the

hypotheses of lemma 67, given a homogeneous

local rule (N

+

, A, S

0

) with neighbourhood sys-

tem N

+

such that |N

+

e

G

| = |S

+

|, the local

rule (N

0+

, A,

˜

S

0

), with

˜

S

0

e

H

:

= S

0

e

G

is well-

deﬁned on the Cayley graph Γ(H, φ(S

+

)), with

N

0+

φ(g)

= φ(N

+

g

). Viceversa, given a local

rule (N

0+

, A,

˜

S

0

) whose neighbourhood system

N

0+

corresponds to a Cayley graph Γ(H, φ(S

+

)),

the local rule (N

+

, A, S

0

), with S

0

e

:

=

˜

S

0

e

is

well-deﬁned on the Cayley graph Γ(G, S

+

), with

φ(N

+

g

) = N

0+

φ(g)

Proof. The ﬁrst part of the proof proceeds ex-

actly as that of lemma 68. The non trivial part

of the proof regards the equivalence of Eq. (110)

for both S

0

on G and

˜

S

0

on H. Suppose that

S

0

deﬁnes a local rule for the neighbourhood sys-

tem N

+

g

. In theories of type 2, it is suﬃcient

to verify that Eq. (110) holds for regions with

|R| = 2, |S| = 0, or |R| = 1, |S| = 1, or |R| = 0,

|S| = 2. Let then R, S ⊆ Γ(H, φ(S

+

)), with

R = φ(

˜

R) and S = φ(

˜

S). In the ﬁrst case we have

R = {f

1

, f

2

}, with

˜

R = {g

1

, g

2

} and f

i

= φ(g

i

),

while S = ∅. The relevant informations in or-

der to verify Eq. (110) are the cardinalities i)

|N

0−

f

1

∩ N

0−

f

2

| and ii) |P

0−

f

1

∩ P

0−

f

2

|. If f

2

∈ P

0−

f

1

,

by lemma 67 we know that we can choose g

1

and g

2

so that |N

0−

f

1

∩ N

0−

f

2

| = |N

−

g

1

∩ N

−

g

2

| and

|P

0−

f

1

∩ P

0−

f

2

| = |P

−

g

1

∩ P

−

g

2

|. Thus, since Eq. (110)

holds for C ∈ [[A

(G)

˜

R

→ A

(G)

˜

R

]], and

A

(G)

˜

R

∼

=

A

(H)

R

,

˜

S

0

= S

0

,

we conclude that Eq. (110) holds also for C ∈

[[A

(H)

R

→ A

(H)

R

]]. On the other hand, if f

2

6∈ P

0−

f

1

,

we can calculate

˜

S

0

N

0−

f

1

,f

2

C

˜

S

0

N

0−

f

1

,f

2

in two steps.

First we apply

˜

S

0

N

0−

f

1

= S

0

N

−

g

1

, treating the system

f

2

as an external system. Reminding that f

2

6∈

P

0−

f

1

, it is N

0−

f

1

∩ N

0−

f

2

= ∅. By Eq. (110), that

holds for S

0

N

−

g

1

we then have

(P

0−

f

2

\[P

0−

f

1

∪f

2

],0)

(f

2

,0)

C

(f

2

,0)

(P

0−

f

1

,0)

˜

S

0

N

0−

f

1

(f

1

,0) (f

1

,0)

˜

S

0

N

−

f

1

(P

0−

f

1

,0)

(P

0−

f

1

\f

1

,0)

(N

0−

f

1

,1) (N

−

f

1

,1) (N

0−

f

1

,1)

(N

0−

f

2

,1)

=

(P

0−

f

1

∪P

0−

f

2

\f

2

,0)

(f

2

,0)

C

0

(f

2

,0)

(N

0−

f

1

,1) (N

0−

f

1

,1)

(N

0−

f

2

,1)

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 53

Then, we apply

˜

S

0

N

0−

f

2

, obtaining

(N

0−

f

1

,1)

C

0

(N

0−

f

1

,1)

(N

0−

f

2

,1)

˜

S

0

N

0−

f

2

(f

2

,0) (f

2

,0)

˜

S

0

N

0−

f

2

(N

0−

f

2

,1)

(N

0−

f

2

,1)

(P

0−

f

2

,0) (P

0−

f

2

\f

2

,0) (P

0−

f

2

,0)

(P

0−

f

1

,0)

(P

0−

f

2

\[P

0−

f

1

∪f

2

],0)

=

(P

0−

f

1

∪P

0−

f

2

,0)

(N

0−

f

1

,1)

C

00

(N

0−

f

1

,1)

(N

0−

f

2

,1) (N

0−

f

2

,1)

, (116)

where on the l.h.s. we reversed the ordering of

the wires for the sake of simplicity of the di-

agram. Overall, considering that

˜

S

0

N

0−

f

1

∪N

0−

f

2

=

˜

S

0

N

0−

f

1

˜

S

0

N

0−

f

2

=

˜

S

0

N

0−

f

2

˜

S

0

N

0−

f

1

, we then proved

Eq. (110) for the local rule

˜

S

0

f

on the neighbour-

hood scheme given by the graph Γ(H, φ(S

+

)).

In the second case, let R = f

1

= φ(g

1

), S =

f

2

= φ(g

2

). The informations that matter are

whether one can choose g

1

, g

2

so that f

2

∈ N

0−

f

1

iﬀ g

2

∈ N

−

g

1

, and |P

0−

f

1

∩N

0+

f

2

| = |P

−

g

1

∩N

+

g

2

|. Again

by lemma 66, we know that the answer is posi-

tive. Then, also for |R| = |S| = 1, since Eq. (110)

holds for C ∈ [[A

(G)

(

˜

R,0)∪(

˜

S,1)

→ A

(G)

(

˜

R,0)∪(

˜

S,1)

]], and

A

(G)

(

˜

R,0)∪(

˜

S,1)

∼

=

A

(H)

(R,0)∪(S,1)

,

˜

S

0

= S

0

,

we conclude that Eq. (110) holds also for C ∈

[[A

(H)

(R,0)∪(S,1)

→ A

(H)

(R,0)∪(S,1)

]]. The argument for

|R| = 0, |S| = 2 is the same as for |R| = 2,

|S| = 0. For the converse, one can follow the

same argument as for the direct statement. The

only diﬀerences are that in the case |R| = 2

and |S| = 0 one might have g

2

6∈ P

−

g

1

while

f

2

= φ(g

2

) ∈ φ(P

−

g

1

) = P

0−

f

2

, and in the case

|R| = |S| = 1 it might happen that g

2

6∈ N

−

g

1

while f

2

∈ N

0−

f

1

. By lemma 67, the ﬁrst case

can occur only if N

+

g

1

∩ N

+

g

2

= P

−

g

1

∩ P

−

g

2

= ∅,

while by lemma 66 the second case occurs only

if P

−

g

1

∩ N

+

g

2

= ∅. In both cases, the diﬀerent

structure of the neighbourhoods between G and

H does not preclude the derivation of Eq. (110)

in the case of G from the same identity for the

case of H, as one can immediately realise by di-

rect inspection of Eq. (110).

Theorem 12 (Wrapping lemma). For a theory

of type 1, under the hypotheses of lemma 66,

the local rule (N

+

, A, S

0

) deﬁnes a cellular au-

tomaton (G, A, V ) on the Cayley graph Γ(G, S

+

)

if and only if the same local rule deﬁnes a cel-

lular automaton (H, A, W ) on the Cayley graph

Γ(H, φ(S

+

)). The same is true of a theory of

type 2, under the hypotheses of lemma 67.

Proof. The thesis now easily follows from lem-

mas 68 and 69, and theorem 10.

The above result allows us, at least in theories

of the two types considered so far, and for CAs

on Cayley graphs of groups with ﬁnite quotients

satisfying the hypotheses of lemmas 66 or 67, to

reduce the classiﬁcation of CAs on graphs of inﬁ-

nite groups to that of CAs on ﬁnite groups, which

is in principle a much easier task. Moreover, once

a suitable ﬁnite quotient is found, one can simu-

late the evolution of a ﬁnite region and for a ﬁnite

number of steps considering the same evolution

on the ﬁnite wrapped graph instead of having to

consider the actual inﬁnite graph.

9 Examples

9.1 Classical case

Classical systems d can be classiﬁed in terms of

the number d of pure states ρ

y

, y = 1, . . . , d which

coincides with the dimension of the state space

d = dim[[d]]

R

. All the pure states of a classical

system are jointly discriminable by atomic eﬀects

a

x

, x = 1, . . . , d, i.e. (a

x

|ρ

y

) = δ

xy

. For the sake

of simplicity, we will restrict attention to classi-

cal computation, i.e. the sub-theory of classical

theory where there is only one kind of non-trivial

elementary system, the bit having d = 2, and all

other systems are composite systems of n bits,

having dimension d = 2

n

. Every eﬀect in [[

¯

A

G

]]

Q

for an inﬁnite system |G| = ∞ can be decom-

posed as a series of eﬀects corresponding to ﬁnite,

arbitrarily large bit strings b

R

, with

b

R

:

= [a

R

], a = a

b

r

1

⊗ a

b

r

2

⊗ . . . ⊗ a

b

r

|R|

,

with R = {r

1

, r

2

, . . . , r

|R|

}. In this case, let b

R

de-

note a local eﬀect. A global rule V can be spec-

iﬁed by the input/output relation V

†

b

R

= b

0

R

0

.

However, in the extensive literature about classi-

cal cellular automata—see e.g. [59] for reference,

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 54

though we cannot even try to provide an exhaus-

tive account of the relevant bibliography—it is

customary to express V through its local rule,

which is speciﬁed by its action on local states.

The only point we want to discuss here is the

apparent clash of our general wrapping lemma

with results in the literature. In particular, in

Ref. [31] a counterexample to the classical ver-

sion of the wrapping lemma is provided, referring

to [48]. The local rule for the counterexample is

the following. Let G = Z

r

with r 6∈ 3Z. The rule

deﬁned by

b

0

j

= b

j

r

1

⊕ b

j

⊕ b

j⊕

r

1

(117)

What we claim here is that, according to our

notion of neighbourhood based on deﬁnition 36,

the above automaton has a neighbourhood sys-

tem where N

+

i

is actually much larger than

{j 

r

1, j, j ⊕

r

1}. Indeed, if one considers the

action of V on the algebra of transformations in-

stead of its action on states, one can easily realise

that N

+

i

= G for every i. However, by checking

the conditions for no-signalling, it is easy to re-

alise that the rule in Eq. (117) only signals to

sites i  1, i, and i ⊕ 1. Thus, while no-signalling

is necessary for no causal inﬂuence, the converse

is not true.

We remark that the deﬁnition of causal inﬂu-

ence that we give, based on transformations, is

inspired by the quantum deﬁnition, which indeed

involves the generators of the local algebra, and

thus the Kraus representatives of transformations

rather than the mere set of local states. In this

respect, the notion of neighbourhood used in the

literature on CA is diﬀerent from the quantum

one, and we point to this relevant diﬀerence as

the responsible for the apparent incompatibility

discussed in Ref. [31]. The subject was exten-

sively studied in Ref. [60] and later in Ref.[61],

where the authors maintain the notion of neigh-

bourhood for classical CA based on the signalling

condition, and prove a bound on the mismatch

between the classical neighbourhood of a given

CA and the neighbourhood of the quantum ver-

sion of the same CA.

9.2 Quantum case

Quantum cellular automata represent now a

widely studied ﬁeld of research. After the very

general analysis of Refs. [31, 32, 33, 61], there are

various results in the literature about quantum

cellular automata. Here we cite the very general

classiﬁcation of Ref. [62, 63], while for extensive

reviews we refer to Refs. [64, 65].

9.3 Fermionic case

For the presentation of fermionic theory as an

OPT we refer to [36, 37]. The Fermionic systems

are fully speciﬁed by the algebra A generated by

local ﬁeld operators ψ

i

. In fact, A is the alge-

bra of ﬁeld operator polynomials. This algebra is

Z

2

-graded, with two modules A

E

and A

O

, given

by the span of even and odd polynomials, respec-

tively. No combination of even and odd poly-

nomials is allowed. As in all quantum theories,

A provides a simpliﬁed representation in terms

of Kraus operators of the algebra of transforma-

tions, which is not graded, but rather reducible

to a direct sum. Linear combinations of transfor-

mations indeed do not correspond to linear com-

binations of the corresponding Kraus operators.

The local systems for a site g of a cellular au-

tomaton in this case are registers made of one

or more local Fermionic modes. Local ﬁeld op-

erators are then labelled ψ

i,g

, where g ∈ G and

i ∈ J

g

= {1, . . . , n

g

}, |J

g

| representing the size

of the register at site g. Since the local algebra

A

g

is ﬁnitely generated, the locality requirement

implies that V is given by a local rule, which in

turn is completely speciﬁed by its action on the

local generators ψ

i,g

for every g ∈ G. Thus, one

has

V (ψ

i,f

)

=

X

g∈(N

+

f

)

×k

j∈J

g

s,t∈{0,1}

∗

T

(g,j,s,t)

i,f

ψ

s

j

1

j

1

,g

1

ψ

†t

j

1

j

1

,g

1

. . . ψ

s

k

j

k

,g

k

ψ

†t

i

k

j

k

,g

k

,

where J

g

:

=

×

g∈N

+

f

J

g

. Most of the automata

studied so far in the literature are linear, namely

V (ψ

i,f

) =

X

g∈N

+

f

j∈J

g

T

j,g

0

i,f

ψ

j,g

.

Remarkable exceptions are represented by

Refs. [44, 66, 45]. In particular, in Ref. [45] the

full classiﬁcation of Fermionic cellular automata

with |J

g

| = 1 on Cayley graphs of Z

2

× Z

2

and Z

was carried out.

As regards Fermionic cellular automata, we

would like to remark that, despite Fermionic

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 55

theory being of type 2, they satisfy the wrap-

ping lemma in the form of lemma 68, instead of

lemma 69. This is due to the fact that Fermionic

transofrmations, like quantum transformations,

admit a Kraus decomposition, and can thus be

represented through Kraus operators instead of

completely positive maps. It turns out that local

Fermionic ﬁeld operators ψ

j,g

and ψ

†

j,g

provide a

basis for the full algebra of Fermionic operators.

Thus, in the Fermionic case, if Eq. (110) is sat-

isﬁed for local transformations, it is always sat-

isﬁed. Real quantum theory, which shares many

features with Fermionic theory, is diﬀerent in this

respect, and we conjecture that a counterexam-

ple to the thesis of lemma 68 can be found in the

latter case.

10 Conclusion

Summarising the content of the manuscript, we

deﬁned the composition of denumerably many

systems from an OPT, starting from the space of

quasi-local eﬀects, and deﬁning quasi-local trans-

formations. We then deﬁne states of such system

in terms of suitable functionals on the space of

eﬀects. Among these, there are some that can

be interpreted as the result of local preparations,

and that we deem quasi-local states.

We deﬁned global update rules for ﬁnite or in-

ﬁnite systems, and proved a generalisation of a

block-decomposition theorem for GURs. We then

analysed those GURs that are homogeneous. The

deﬁnition of homogeneity is a big chapter on its

own, and deserves a careful treatment. As a con-

sequence of the deﬁnition, we proved that for ho-

mogenous GURs the memory array is organised

as the Cayley graph of a group. We then deﬁned

locality for GURs, and studied its consequences

in detail.

In the last section we used homogeneity and lo-

cality to deﬁne cellular automata, and proved the

wrapping lemma under very general assumptions.

The theory developed here is of fundamental

importance for an approach to physical laws as

information-processing algorithms, and will be

used for the construction of statistical mechanics

in post-quantum theories, as well as the recon-

struction of dynamical laws in space-time beyond

the now established approach based on quantum

walks, thus encompassing interacting quantum

and post-quantum ﬁeld theories.

Acknowledgments

This publication was made possible through the

support of a grant from the John Templeton

Foundation under the project ID#60609 Causal

Quantum Structures. The opinions expressed

in this publication are those of the authors and

do not necessarily reﬂect the views of the John

Templeton Foundation. The author wishes to

thank G. M. D’Ariano, A. Bisio, A. Tosini, and

F. Buscemi for many inspiring discussions. Sug-

gestions about the terminology by G. Chiribella

are also acknowledged that were useful to sharpen

the notion that is now deﬁned as causal inﬂuence.

A particular thank to J. Barrett for fruitful con-

versations about various facets of causal inﬂuence

and signalling, and to M. Erba for his help in tun-

ing the deﬁnitions and dissipating misconceptions

on the same subject.

A Identiﬁcation of the sup- and oper-

ational norm for eﬀects

We show that, under the hypothesis of assump-

tion (16), one has kak

op

= kak

sup

for every

a ∈ [[

¯

A]]

R

. Indeed, if [[

¯

A]]

+

= [[A]]

∗

+

, then, by

theorem 5, one has

[[

¯

A]] = {a ∈ [[

¯

A]]

R

| 0 ≤ (a|ρ), (e − a|ρ), ∀ρ ∈ [[A]]}.

(118)

First of all, one can rephrase the above condition

as

[[

¯

A]] = {a ∈ [[

¯

A]]

R

| 0 ≤ (a|ρ), (e − a|ρ), ∀ρ ∈ [[A]]

1

}.

(119)

Indeed, since [[A]]

1

⊆ [[A]], the set on the l.h.s. in

Eq. (118) is clearly a subset of that on the l.h.s. in

Eq. (119). Moreover, since every state ρ ∈ [[A]]

is proportional by a constant κ ∈ [0, 1] to a

deterministic one ρ

0

∈ [[A]]

1

(see remark 1), if

0 ≤ (a|σ), (e − a|σ) for all σ ∈ [[A]]

1

, then

0 ≤ (a|ρ), (e − a|ρ) for all ρ ∈ [[A]]. Then, by

the identity in Eq. (119), one can identify the set

[[

¯

A]] as the subset of [[

¯

A]]

R

containing only those

functionals a on [[A]]

R

such that 0 ≤ (a|ρ) and

0 ≤ 1 − (a|ρ) for ρ ∈ [[A]]

1

, and, in turn, the

above condition is equivalent to

[[

¯

A]] = {a ∈ [[

¯

A]]

R

| 0 ≤ (a|ρ) ≤ 1, ∀ρ ∈ [[A]]}.

Now, we remind that λ ∈ J(a) if and only if

λe ± a  0. Under our hypotheses, this means

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 56

that λ ± (a|ρ) ≥ 0 for every ρ ∈ [[A]]. Thus,

|(a|ρ)| ≤ λ for every ρ ∈ [[A]]. Finally, we can con-

clude that kak

op

≤ kak

sup

. On the other hand, if

kak

op

< kak

sup

, then there exists ε > 0 such that

for every ρ ∈ [[A]] one has |(a|ρ)| ≤ kak

sup

− ε.

In particular, this is true for ρ

1

∈ [[A]]

1

. Conse-

quently

|(a|ρ)| = κ|(a|ρ

1

)| ≤ κ(kak

sup

− ε) ∀ρ ∈ [[A]],

where 0 ≤ κ ≤ 1 and ρ = κρ

1

. Then (e|ρ) = κ,

and

({kak

sup

− ε}e ± a|ρ) ≥ 0, ∀ρ ∈ [[A]].

Finally, this implies that kak

sup

−ε ∈ J(a), which

is absurd. Then, kak

sup

= kak

op

.

B Quasi-local states

We start considering arbitrary ﬁnite disjoint par-

titions of the system A

G

. Let B

:

= {B

i

}

∞

i=1

de-

note a disjoint partition of the set G into ﬁnite

blocks, i.e. 0 < |B

i

| < ∞ for every i ∈ N. The

set of such partitions (quotiented by the irrele-

vant ordering of regions) is denoted by B.

Deﬁnition 51. For a given partition B ∈ B, we

deﬁne the set of ﬁnite B-regions of G as

R

(G,B)

:

=







R ⊆ G | R =

k

[

j=1

B

i

j

; 0 ≤ k < ∞







,

(120)

and the set of B-regions of G as

R

(G,B)

:

=







R ⊆ G | R =

k

[

j=1

B

i

j

; 0 ≤ k ≤ ∞







,

(121)

where for k = 0 we deﬁne

0

[

j=1

B

i

j

:

= ∅.

Given R =

S

k

j=1

B

i

j

in R

(G,B)

, let N

(B)

R

:

=

{i

j

}

k

j=1

.

Remark 10. Clearly, R

(G,B)

⊆ R

(G,B)

. It is also

trivial to prove that R

(G,B)

and R

(G,B)

are closed

under ∪, ∩, \.

Given a ﬁnite region R ∈ R

(G)

, for any parti-

tion B ∈ B it is possible to cover R with regions

of B.

Deﬁnition 52. Given B ∈ B, the cover in B of

a region R ⊆ G, denoted B(R), is the minimal

B-region C =

S

k

j=1

B

i

j

such that R ⊆ C.

The above notion of cover is well deﬁned, as

proved by the following lemma.

Lemma 70. Let R ⊆ G. The cover B(R) in B

of R exists and is unique.

Proof. First of all, the set of B-regions S ∈

R

(G,B)

such that R ⊆ S is not empty, since

R ⊆ G ∈ R

(G,B)

. We just need to prove that,

having deﬁned C(R, B)

:

= {S ∈ R

(G,B)

| R ⊆ S},

there is always a unique minimal element B(R) ∈

C(R, B), i.e. such that no B-region strictly con-

tained in B(R) covers R. As to existence, let

h ∈ R. Then h ∈ B

l

h

for some l

h

. One can eas-

ily realise that R ⊆ C

:

=

S

h∈R

B

l

h

. Moreover,

if we remove a given B-region B

l

h

from C, and

h ∈ B

l

h

, then h 6∈ C \ B

l

h

, thus C is a minimal

cover of R in B. As to uniqueness, suppose that

there are two diﬀerent minimal covers of R in B,

say C

1

and C

2

. In this case, it is not restrictive

to suppose that there is h ∈ G such that h ∈ C

1

and h 6∈ C

2

. Let B

i

∈ B be the set such that

h ∈ B

i

. One has B

i

⊆ C

1

and B

i

∩ C

2

= ∅. Since

R ⊆ C

2

, we can conclude that R ∩ B

i

= ∅, and

thus R ⊆ (C

1

\ B

i

) ∈ C(R, B). Thus, C

1

cannot

be minimal in C(R, B).

The cover B(R) can also be thought of as the

intersection of all covers of R in B.

Lemma 71. The cover B(R) of a ﬁnite region

R is ﬁnite.

Proof. Every element g

k

∈ R belongs to some

B

i

k

⊆ B, where k 7→ i

k

is generally not injec-

tive. It is then clear that R ⊆

S

|R|

k=1

B

i

k

, and

thus B(R) ⊆

S

|R|

k=1

B

i

k

.

Given one partition B ∈ B, we can now deﬁne

the set of states that diﬀer on ﬁnitely many blocks

from a given assignment of states of blocks in B.

First, we deﬁne the reference state as follows.

Deﬁnition 53 (Reference local state). Given a

partition B ∈ B, a reference local state over B is

a map

ρ

0

: N →

G

j∈N

[[A

B

j

]]

1

:: j 7→ ρ

0j

∈ [[A

B

j

]]

1

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 57

A reference local state allows one to attribute a

state to any ﬁnite region R ∈ R

(G)

by the follow-

ing procedure. First, let us consider the B-cover

B(R) of R. Then, the state of B(R) is given by

ρ

0

[B(R)]

:

=

O

i

j

∈N

(B)

B(R)

ρ

0i

j

.

Finally, we discard all the extra systems in the

region B(R) \ R.

ρ

0|R

:

= ρ

0

[B(R)]

|R

.

The above construction is then formalised in

the following deﬁnition.

Deﬁnition 54. The state of a ﬁnite region R ∈

R

(G,B)

in the reference local state ρ

0

is deﬁned as

ρ

0|R

R

:

=

ρ

0

[B(R)]

R

B(R)\R

e

,

ρ

0|∅

:

= 1.

Given a partition B and a reference local state

ρ

0

, let us now deﬁne the set of primitive local

states as follows.

Deﬁnition 55. The set of primitive local states

is

Pre[[A

G

]]

(B)

R

:

= {(σ, R) | σ ∈ [[A

R

]]

R

; R ∈ R

(G,B)

}.

This set collects all the generalised local prepa-

rations of ﬁnite regions of B. As in the case

of eﬀects, we introduce the simpliﬁed notation

σ

S

:

= (σ, S). A choice of reference local state

ρ

0

, allows us to interpret a given element σ

S

of

Pre[[A

G

]]

(B)

R

as a preparation of the region S in

the state σ. Thus, the element σ

S

allows one to

attribute a generalised state τ ∈ [[A

T

]]

R

to any

region T ∈ R

(G)

as follows (see ﬁg. 3)

τ

:

= σ

|S∩T

⊗ ρ

0|(T \S)

. (122)

Deﬁnition 56. Given a reference local state ρ

0

over B we deﬁne the following equivalence rela-

tion in Pre[[A

G

]]

(B)

R

σ

S

∼

ρ

0

τ

T

⇔

(

σ = ν ⊗ ρ

0|(S\T )

,

τ = ν ⊗ ρ

0|(T \S)

,

(123)

for some ν ∈ [[A

S∩T

]]

R

.

⇢

01

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⇢

02

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⇢

03

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⇢

04

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⇢

05

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⇢

06

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⇢

07

<latexit sha1_base64="nLKtHiRE5+IoXsYlmAr3AuR3yRM=">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</latexit>

⇢

08

<latexit sha1_base64="wxVEbPaVRzCK3XoX3MzVgBUPV88=">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</latexit>

⇢

09

<latexit sha1_base64="VlqwQbJd7gn9CLhEDH5wkqDeAMQ=">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</latexit>

⇢

010

<latexit sha1_base64="77XfSYRmbYpF2Wa0MwevPeH7A6U=">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</latexit>

⇢

011

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B

1

<latexit sha1_base64="hdKGxyj9oDmC260d3KDDH1MX91I=">AAAIT3icdZVbb9s2FMfV7uI0u7Xr417YqQU6IDAsJ+sFCILGjnNperEtO3FSBQFJUY6ga0i6iKHqI+x1+1x73CfZ2zBKJMV6awUE4fmd/7nwyKRQHoeMdzp/3br9xZdffd1au7P+zbffff/D3Xs/nrBsQTGZ4izO6AxBRuIwJVMe8pjMckpggmJyiqJ+5T99TygLs3TClzm5SOA8DYMQQy6Q27t0Lu/anfZm99dnziaQiydbcvH0+RZw2p36sS31DC/vtZ54foYXCUk5jiFj75xOzi8KSHmIY1KuewtGcogjOCdF3WEJHgnkgyCj4i/loKYrOpiwBPKrDfGfLRO0wXgC6ZL6GygR0ZWLregrQlnAyk9l+VTEuwUPnl0UYZovOEmxbClYxIBnoBoM8ENKMI+XYgExDcVeAL6CFGIuxrdSZU5hfhXim3L90UpLS5/UDdU1wM0SeDfLLK/mXMA41g6hG+GQ4kXIBRIhHsux7RSFV/ccANspteOwqJzgUNlDmmWB0BGfRWHu5ZB6aRamvngRhYcCUPvbQIdfE7946G14KBYdsusFpORhFVQC6e8neXwjyyIE+ipqp/AoTOf1i6zs7cKLpV2bUVR4O94D4O1IG6HC2/YeeNslqG1OUxbQRGxI7gfTakO1K07/46RJ5ZNxdFdMQIDdUoppT9o9vRval6CvBXvS3iv1fuhAkkGpc+5LsG8kB5IcNJIjCY6aMlgCrMuMpD0yOcaSjJscrgSukcwkmTWSMwnOjORckvOG8N1CT09MQaKeQT3VIO8b1leyPYP2mnSuga7SHRh0oNO9MuyVkr0x6I2WHRt2rF/1kWFHWjcxbKLSTQ2aNt2dGHiidKcGnep0rw17rWQzg2ZadmbYmZKdG3SuZfuGiR+FZG8Ne6t1A8MGerMvP4p9qYVjA8cq39CgISg//y5WBqVPZMoZ4c1VUAyCoGx+H+6qz+XGNRGJPvZNxAlmVagqPzK1RorNaV5fOiiL/erGzeKiunjUbt0cpqaSMHSqiIuoD7Yjjj+QUkSr47xtOx+UxI+UJIq0xkeVBiEjOpaaX7SiJ+zHxu32bcfu1sTuNqIJlWdmQpsbEsNaWNe3u6ar/0sHHFakOFQHWwysHsDKjTsmMGZFcyeONUfvibyjUSDHpLAYUuUCqQaRApEG1wpcl+viS6s/p+Dzi5Nu29lsd0db9oue+uauWT9ZP1uPLcd6ar2wDq2hNbWwNbd+s363/mj92fq79c+akt6+pRb3rZVn7c6/pHjroQ==</latexit>

B

2

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B

3

<latexit sha1_base64="SIpjGbzmQ5IqgvG0T7gt7KKi62s=">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</latexit>

B

4

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B

9

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B

10

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B

11

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⇢

01

<latexit sha1_base64="wcn3oa0+0vL9+3GXSv1gdPQl2qo=">AAAIVXicdZVpb9s2GMfVY42bHT32sm/YqQU6IDAsJz0GBEFjxzmaHj4TO2UQUBTlCDpD0kUMVZ9ib7fPNezDDBglkmK9tQIM6/k9/+fgI5FysyhgvNX6+8bNW7e/u7PWuLv+/Q8//nTv/oOHJyxdUEwmOI1SOnURI1GQkAkPeESmGSUodiNy6obd0n/6iVAWpMmYLzNyHqN5EvgBRlygGaSX6UXecoqL+3arudl+/srZBPLmxZa8efnbFnCareqyLXX1Lx6svYBeihcxSTiOEGMfnVbGz3NEeYAjUqzDBSMZwiGak7zqswBPBfKAn1LxSzio6IoOxSxG/HJD/LNl7G4wHiO6pN6GG4vo0sVW9CWhzGfF17J8LeLjgvuvzvMgyRacJFi25C8iwFNQjgd4ASWYR0txgzANxFoAvkQUYS6GuFJlTlF2GeDrYv3pSktLj1QNVTXA9RLA62WaldPOURRph9ANcEDxIuACiRDIMmw7eQ6rnn1gO4V2HOalExwqu0/T1Bc64rEwyGCGKEzSIPHEg8ih64PK3wQ6/Ip4+RO4Ad1IdMiuFoiSJ2VQAaS/G2fRtSzruqCronZySFEyrx5kaW/nMJJ2ZYZhDnfgYwB3pO26OdyGj+F2ASqb04T5NBYLkuvBtFxQ5YqS/zhpXPpkHN0VExBgt5Bi2pF2R6+GdiXoasGetPcKvR7ak6RX6Jz7EuwbyYEkB7XkSIKjugyWAOsyA2kPTI6hJMM6x0iCkZFMJZnWkpkEMyM5k+SsJnw319MTU5CoY1BHNci7hnWVbM+gvTrdyMCR0h0YdKDTvTXsrZK9N+i9lh0bdqwf9ZFhR1o3Nmys0k0MmtTdnRh4onSnBp3qdO8Me6dkU4OmWjYzbKZkZwadadm+YeKlkOyDYR+0rmdYTy/2zRexb7RwaOBQ5esb1AfFt5/FyqD0jkw4I7w+CvKe7xf1+zFa9Y24cY1Foi99Y7GDWRmqyg9MrYFic5pVh46bRl554qZRXh48arWjDCWmkjB0qpCLqM+2I7Y/kFKXltt523Y+K4kXKkkYao3nlhrXNaJjqflVKzrCfmbco67t2O2K2O1aNKZyz4xpfUJiVAmr+nbbdPV/aY+jkuSHamOLgVUDWDlxhwRFLK/PxKHm7iciz2jXl2NSWAypdIFEg1CBUIMrBa6KdfGl1Z9T8O2bk3bT2Wy2B1v264765jasR9Yv1jPLsV5ar61Dq29NLGzF1u/WH9afa3+t/dO43bgjpTdvqJifrZWrce9fsXrtaQ==</latexit>

⇢

02

<latexit sha1_base64="k9GxzKcL0UbtRm+kbfu6qag/vDA=">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</latexit>

⇢

03

<latexit sha1_base64="sOt3vdlSUKaja88vDY1pstla2yI=">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</latexit>

⇢

06

<latexit sha1_base64="AUdkV9uV3TzC83+guFueH3YQvEQ=">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</latexit>

⇢

09

<latexit sha1_base64="VlqwQbJd7gn9CLhEDH5wkqDeAMQ=">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</latexit>

⇢

010

<latexit sha1_base64="77XfSYRmbYpF2Wa0MwevPeH7A6U=">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</latexit>

⇢

011

<latexit sha1_base64="bXSZkKN7UVRmN000ftK0y9OLlN4=">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</latexit>

B

1

<latexit sha1_base64="hdKGxyj9oDmC260d3KDDH1MX91I=">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</latexit>

B

2

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B

3

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B

6

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B

9

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B

10

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B

11

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B

4

[ B

5

[ B

7

[ B

8

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

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(b)

Figure 3: An illustrative example of a reference local

state ρ

0

(a) for a partition of a ﬁnite set G into 11

regions, along with a primitive local state σ

R

(b) with

R = B

4

∪B

5

∪B

7

∪B

8

. The dots represent the elements

of G.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 58

The symbol σ

Sρ

0

will denote the equivalence

class of σ

S

∈ Pre[[A

G

]]

(B)

R

under the relation ∼

ρ

0

.

Notice that in Eq. (123), thanks to closure of

R

(G,B)

under set diﬀerence, one has (S \ T ), (T \

S) ∈ R

(G,B)

, and thus the deﬁnition of the equiva-

lence relation ∼

ρ

0

involves exclusively B-regions.

Moreover, all the regions involved in the deﬁni-

tion are ﬁnite.

Lemma 72. Let σ

S

∈ Pre[[A

G

]]

(B)

R

. Then for

every R ∈ R

(G,B)

such that R∩S = ∅ one has (σ⊗

ρ

0|R

)

S∪R

∈ Pre[[A

G

]]

(B)

R

and (σ ⊗ ρ

0|R

)

S∪R

∼

ρ

0

σ

S

.

Proof. First of all, as R

(G,B)

is closed under

union, clearly S ∪ R ∈ R

(G,B)

, and thus (σ ⊗

ρ

0|R

)

S∪R

∈ Pre[[A

G

]]

(B)

R

. Now, S ∩ (S ∪ R) = S,

and thus the condition in Eq. (123) is satisﬁed

for T = S ∪ R, T \ S = R, S ∩ T = S, ν = σ,

ρ

0|(T \S)

= ρ

0|R

, and ρ

0|(S\T )

= 1.

We now provide a way to identify a canonical

representative of the equivalence class σ

Rρ

0

de-

ﬁned as follows.

Deﬁnition 57. The minimal representative ˜σ

R

σ

of the equivalence class σ

Rρ

0

is deﬁned through

R

σ

:

=

\

S∈R

(σ,R)

S, ˜σ

R

σ

∼

ρ

0

σ

R

, (124)

where

R

(σ,R)

:

= {S ∈ R

(G,B)

| ∃τ ∈ [[A

S

]]

R

: τ

S

∼

ρ

0

σ

R

}.

Lemma 73. The minimal representative exists

and is unique.

For the proof, see the analogous lemma 9.

Deﬁnition 58. The set of generalised local

states over B upon ρ

0

is

[[A

G

]]

(B)

ρ

0

LR

:

= Pre[[A

G

]]

(B)

R

/ ∼

ρ

0

. (125)

Its elements will be denoted by symbols like ρ

Sρ

0

.

We now deﬁne linear combinations of local

states.

Deﬁnition 59. Let σ

Sρ

0

, τ

T ρ

0

∈ [[A

G

]]

(B)

ρ

0

LR

, and

a ∈ R. Then we have

a(σ

Sρ

0

)

:

=

(

(aσ)

Sρ

0

a 6= 0

0

∅ρ

0

a = 0,

σ

Sρ

0

+ τ

T ρ

0

:

= ν

S∪T ρ

0

,

ν

:

= σ ⊗ ρ

0|(T \S)

+ ρ

0|(S\T )

⊗ τ.

We will denote by R

σ+τ

the region where the

minimal representative of the class σ

Sρ

0

+ τ

T ρ

0

diﬀers from ρ

0

. As in the case of eﬀects, it is not

always true that R

σ+τ

= R

σ

∪R

τ

. As an example,

consider σ = α

i

1

⊗ β

i

2

and τ = α

i

1

⊗ (ρ

0i

2

− β

i

2

),

with α

i

1

6= ρ

0i

1

, β

i

2

6= ρ

0i

2

, and R

σ

= R

τ

=

B

i

1

∪ B

i

2

. Then σ + τ = α

i

1

⊗ ρ

0i

2

, and clearly

R

σ+τ

= B

i

1

, which is strictly included in R

σ

=

R

τ

= R

σ

∪ R

τ

.

Deﬁnition 60. The set of local states over B

upon ρ

0

is

[[A

G

]]

(B)

ρ

0

L

:

= {σ

Sρ

0

∈ [[A

G

]]

(B)

ρ

0

LR

| σ ∈ [[A

S

]]}.

An element σ

Sρ

0

∈ [[A

G

]]

(B)

ρ

0

LR

is a local state over

B upon ρ

0

. A subset of special interest is that of

deterministic local states over B upon ρ

0

, deﬁned

as

[[A

G

]]

(B)

ρ

0

L1

:

= {σ

Sρ

0

∈ [[A

G

]]

(B)

ρ

0

L

| σ ∈ [[A

S

]]

1

}.

We now characterise the vector space of gener-

alised states as follows.

Lemma 74. The set of generalized local states

in B upon ρ

0

equipped with the operations deﬁned

in Def. 59 is a real vector space. It is the space

of ﬁnite real combinations of deterministic local

states in B upon the same state ρ

0

:

[[A

G

]]

(B)

ρ

0

LR

= Span

R

([[A

G

]]

(B)

ρ

0

L1

), (126)

The proof is trivial and we omit it.

We now show that quasi-local states are

bounded linear functionals on [[

¯

A

G

]]

QR

. We start

considering the pairing of local states [[A

G

]]

(B)

ρ

0

LR

with local eﬀects [[

¯

A

G

]]

LR

, deﬁned as follows.

Deﬁnition 61. Every σ

Sρ

0

∈ [[A

G

]]

ρ

0

LR

iden-

tiﬁes a functional on [[

¯

A

G

]]

LR

as follows. For

a

R

∈ [[

¯

A

G

]]

LR

let

(a

R

|σ

Sρ

0

)

:

=

σ

|R∩S

R∩S

a

ρ

0|R\S

R\S

. (127)

We now prove that the above pairing is well-

deﬁned, namely it does not depend on the repre-

sentative in the class σ

Sρ

0

, nor on the representa-

tive in the class a

R

. This is done in the next two

lemmas.

Lemma 75. For every a

R

∈ [[

¯

A

G

]]

LR

the result

of Eq. (127) is independent of the choice of rep-

resentative τ

T ρ

0

in the class σ

Sρ

0

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 59

Proof. By hypothesis, there exists ν ∈ [[A

S∩T

]]

R

such that

σ = ν ⊗ ρ

0S\T

, τ = ν ⊗ ρ

0T \S

.

Then we have

(a

R

|σ

Sρ

0

) =(a|ν

|(S∩T )∩R

⊗ ρ

0|[(S\T )∩R]∪(R\S)

)

(a

R

|τ

T ρ

0

) =(a|ν

|(S∩T )∩R

⊗ ρ

0|[(T \S)∩R]∪(R\T )

).

A straightforward set-theoretic calculation gives

that [(S\T )∩R]∪(R\S) = [(T \S)∩R]∪(R\T ) =

R \ (S ∩ T ), and thus (a

R

|σ

Sρ

0

) = (a

R

|τ

T ρ

0

).

On similar lines, we have the following result.

Lemma 76. For every partition B ∈ B, every

reference local state ρ

0

over B, and every local

state τ

T ρ

0

∈ [[A

G

]]

(B)

ρ

0

LR

, the result of Eq. (127) is

independent of the choice of representative b

S

in

the class a

R

.

Proof. If ˜c

R

c

is the minimal representative of

a

R

= b

S

, by Eq. (10), one has

a = ˜c ⊗ e

R\R

c

,

b = ˜c ⊗ e

S\R

c

.

Now, by the deﬁning equation 127, we have that

(a

R

|τ

T ρ

0

) = (a|τ

|R∩T

⊗ ρ

0|(R\T )

)

=(˜c ⊗ e

R\R

c

|τ

|R∩T

⊗ ρ

0|(R\T )

)

=(˜c|τ

|R

c

∩T

⊗ ρ

0|R

c

\T

),

and similarly we obtain

(b

S

|τ

T ρ

0

) = (˜c|τ

|R

c

∩T

⊗ ρ

0|(R

c

\T )

).

The above results give us a hint that for ev-

ery B ∈ B and every ρ

0

, the space [[A

G

]]

(B)

ρ

0

LR

is a

submanifold of the space [[A

G

]]

R

. We now com-

plete the real vector spaces [[A

G

]]

(B)

ρ

0

LR

to Banach

spaces. For this purpose, we start equipping all

spaces [[A

G

]]

(B)

ρ

0

LR

with a norm.

Deﬁnition 62. Given an element σ of the space

[[A

G

]]

(B)

ρ

0

LR

, its norm is deﬁned as

kσk

B,ρ

0

:

= k˜σk

op

, (128)

where k·k

op

denotes the operational norm on

[[A

R

σ

]]

R

.

Lemma 77. Given an element σ of [[A

G

]]

(B)

ρ

0

LR

,

for τ

T

in the corresponding equivalence class, one

has

kσk

B,ρ

0

= kτk

op

,

where k·k

op

denotes the operational norm on

[[A

T

]]

R

.

Proof. By deﬁnition of the relation ∼

ρ

0

, and of

minimal representative ˜σ

R

σ

of σ, one has

τ = ˜σ ⊗ ρ

0(T \R

σ

)

.

Thus, kτ k

op

= k˜σ ⊗ ρ

0(T \R

σ

)

k

op

. Now, by

lemma 3

kτk

op

= k˜σk

op

= kσk

B,ρ

0

.

We can now show that vectors in [[A

G

]]

(B)

ρ

0

LR

are

bounded linear functionals on [[

¯

A

G

]]

QR

.

Deﬁnition 63. Let a = [a

n

R

n

] ∈ [[

¯

A

G

]]

QR

, and

ρ = ρ

Sρ

0

∈ [[A

G

]]

ρ

0

LR

. We deﬁne

(a|ρ)

:

= lim

n→∞

(a

n

R

n

|ρ

Sρ

0

). (129)

With this deﬁnition, we can now prove that lo-

cal states are actually bounded linear functionals

on [[

¯

A

G

]]

QR

.

Lemma 78. For a = [a

n

R

n

] ∈ [[

¯

A

G

]]

QR

, and

ρ

Sρ

0

∈ [[A

G

]]

ρ

0

LR

, one has

|(a|ρ)| ≤ kak

sup

kρk

B,ρ

0

. (130)

Proof. We start considering the case of a local

eﬀect a

R

. In this case, us deﬁne

a

0

:

= (2kak

sup

)

−1

(kak

sup

e

R

+ a)  0,

a

1

:

= (2kak

sup

)

−1

(kak

sup

e

R

− a)  0,

a

0

+ a

1

= e

R

, kak

sup

(a

0

− a

1

) = a,

we have that

|(a

R

|ρ

Sρ

0

)| = |(a

R

|ρ

|R∩S

⊗ ρ

0|R\S

)|

= kak

sup

|({¯a

0

− ¯a

1

} ⊗ e

S\R

|ρ ⊗ ρ

0|R\S

)|

≤ kak

sup

kρ

Sρ

0

k

B,ρ

0

,

where (¯a

i

|

:

= (a

i

|I

R∩S

⊗ |ρ

0|R\S

). Let now a =

[a

n

R

n

]. Then we have

|(a

n

R

n

− a

m

R

m

|ρ

Sρ

0

)|

≤ ka

n

R

n

− a

m

R

m

k

sup

kρ

Sρ

0

k

B,ρ

0

,

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 60

and thus actually (a

n

R

n

|ρ

Sρ

0

) is a Cauchy se-

quence. Finally, since for every n ∈ N one has

|(a

n

R

n

|ρ

Sρ

0

)| ≤ ka

n

k

sup

kρ

S

k

B,ρ

0

,

taking the limit for n → ∞ on both sides we have

that

|(a|ρ

Sρ

0

)| ≤ kak

sup

kρ

Sρ

0

k

B,ρ

0

.

We now show that the deﬁnition in Eq. (129)

is independent of the sequence a

n

R

n

in the class

of a.

Lemma 79. Let a = [a

n

R

n

] = [b

n

R

0

n

] ∈ [[

¯

A

G

]]

QR

,

and ρ

Sρ

0

∈ [[A

G

]]

ρ

0

LR

. Then

lim

n→∞

(a

n

|ρ

Sρ

0

) = lim

n→∞

(b

n

|ρ

Sρ

0

). (131)

Proof. Remind that by deﬁnition, for every ε

there exists n

0

such that for n ≥ n

0

, it is

ka

n

R

n

− b

n

R

0

n

k

sup

≤ ε. Now, by lemma 78, this

implies that

|(a

n

R

n

− b

n

R

0

n

|ρ

Sρ

0

)| ≤ εkρ

Sρ

0

k

B,ρ

0

,

and thus the thesis follows.

We ﬁnally close the vector space [[A

G

]]

(B)

ρ

0

LR

by

adding the limits of Cauchy sequences in the

norm k·k

B,ρ

0

, thus obtaining a Banach space

that we denote [[A

G

]]

(B)

ρ

0

QR

. We start considering

Cauchy sequences

σ : N → [[A

G

]]

(B)

ρ

0

R

:: n 7→ σ

n

R

n

ρ

0

∈ [[A

G

]]

ρ

0

LR

,

which make a real vector space [[A

G

]]

(B)

ρ

0

CR

, con-

taining [[A

G

]]

(B)

ρ

0

LR

as the subspace of constant

sequences. We then quotient [[A

G

]]

(B)

ρ

0

CR

by the

usual equivalence relation

∼

=

, obtaining complete

normed real vector space.

Deﬁnition 64. The space of quasi-local states

in B upon ρ

0

is deﬁned as the space [[A

G

]]

(B)

ρ

0

QR

:

=

[[A

G

]]

(B)

ρ

0

CR

/

∼

=

.

Inside the new Banach space that we deﬁned,

we can single out a convex set of quasi-local

states, corresponding to preparations that can be

arbitrarily approximated by local protocols start-

ing from ρ

0

, along with the cone that they span,

and the convex set of deterministic states.

Deﬁnition 65. An element σ of [[A

G

]]

(B)

ρ

0

QR

is a

quasi-local state if there exists σ

n

R

n

ρ

0

in the class

σ, and n

0

∈ N, such that, for n ≥ n

0

, σ

n

R

n

ρ

0

∈

[[A

G

]]

(B)

ρ

0

L

. The set of quasi-local states in B upon

ρ

0

is denoted by [[A

G

]]

(B)

ρ

0

Q

.

A quasi-local state σ in [[A

G

]]

(B)

ρ

0

Q

is determinis-

tic, if there exists a sequence σ

n

R

n

in the class of

σ and n

0

∈ N such that, for n ≥ n

0

, σ

n

R

n

∈

[[A

G

]]

(B)

ρ

0

L1

. The set of deterministic quasi-local

states is denoted by [[A

G

]]

(B)

ρ

0

Q1

.

We denote the subset of [[A

G

]]

(B)

ρ

0

QR

of elements

σ = λρ, for λ ≥ 0 and ρ ∈ [[A

G

]]

(B)

ρ

0

Q

, as

[[A

G

]]

(B)

ρ

0

Q+

. We will equivalently write σ  0 for

σ ∈ [[A

G

]]

(B)

ρ

0

Q+

Analogously to the case of eﬀects, the ele-

ments of [[A

G

]]

(B)

ρ

0

QR

will be denoted by ρ =

[ρ

n

R

n

ρ

0

], with the convention that for a con-

stant Cauchy sequence ρ

n

R

n

ρ

0

= ρ

Rρ

0

we will

set ρ

Rρ

0

:

= [ρ

n

R

n

ρ

0

]. We deﬁne kρk

B,ρ

0

:

=

lim

n→∞

kρ

n

R

n

ρ

0

k

B,ρ

0

. One can straightforwardly

check that the space [[A

G

]]

(B)

ρ

0

QR

of quasi-local

states in B upon ρ

0

is a Banach space, the sets

[[A

G

]]

(B)

ρ

0

Q

and [[A

G

]]

(B)

ρ

0

Q1

are convex subsets, and

[[A

G

]]

(B)

ρ

0

Q+

is a convex cone. The space [[A

G

]]

(B)

ρ

0

LR

can be identiﬁed with the dense submanifold con-

taining constant sequences σ

Sρ

0

.

For every partition B and every reference local

state ρ

0

, the Banach space [[A

G

]]

(B)

ρ

0

QR

is separa-

ble. Indeed, one can choose a basis in [[A

S

n

]]

R

for each region S

n

:

=

S

n

j=1

B

j

in the increasing

sequence {S

n

}

n∈N

of regions of B. Now, for ev-

ery ρ ∈ [[A

G

]]

(B)

ρ

0

QR

, and for every n ∈ N, given

the class ρ

n

R

n

ρ

0

deﬁning ρ, one has R

n

⊆ S

m

for

suﬃciently large m. Thus, linear combinations of

the countable collection of bases for S

m

are dense

in [[A

G

]]

(B)

ρ

0

QR

.

We now show that actually, for every pair

(B, ρ

0

), the space [[A

G

]]

(B)

ρ

0

QR

is a subspace of

[[A

G

]]

R

, and k·k

∗

≡ k·k

B,ρ

0

on [[A

G

]]

(B)

ρ

0

QR

. First

of all, we extend the pairing of [[

¯

A

G

]]

QR

and

[[A

G

]]

(B)

ρ

0

LR

to [[

¯

A

G

]]

QR

and [[A

G

]]

(B)

ρ

0

QR

.

Deﬁnition 66. Let a = [a

n

R

n

] ∈ [[

¯

A

G

]]

QR

and

σ = [σ

n

S

n

ρ

0

] ∈ [[A

G

]]

(B)

ρ

0

QR

. Then we deﬁne

(a|σ)

:

= lim

n→∞

(a|σ

n

S

m

ρ

0

). (132)

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 61

The above deﬁnition is well-posed, as shown by

the following lemma, whose proof is a straightfor-

ward consequence of lemma 78.

Lemma 80. Let a ∈ [[

¯

A

G

]]

QR

, and σ =

[σ

n

S

n

ρ

0

] = [σ

0

n

S

0

n

ρ

0

] ∈ [[A

G

]]

(B)

ρ

0

QR

. Then

lim

n→∞

(a|σ

m

S

m

ρ

0

) = lim

n→∞

(a|σ

0

m

S

0

m

ρ

0

). (133)

Finally, we use the deﬁnition of sup-norm of ef-

fects to prove that vectors in the spaces [[A

G

]]

(B)

ρ

0

QR

are bounded linear functionals on [[

¯

A

G

]]

QR

. Also

in this case the proof follows straightforwardly

form lemma 78.

Lemma 81. Let a ∈ [[

¯

A

G

]]

QR

and ρ ∈ [[A

G

]]

(B)

ρ

0

QR

.

Then one has

|(a|ρ)| ≤ kak

sup

kρk

B,ρ

0

. (134)

Lemma 82. The spaces [[A

G

]]

(B)

ρ

0

QR

are closed

subspaces of the extended space of states [[A

G

]]

R

.

On [[A

G

]]

(B)

ρ

0

QR

it holds that k·k

B,ρ

0

≡ k·k

∗

.

Proof. We only need to prove that k·k

∗

≡ k·k

Q

,

since we already proved that [[A

G

]]

(B)

ρ

0

LR

is a lin-

ear manifold in the space of bounded linear func-

tionals on [[

¯

A

G

]]

QR

. First of all, by eq. (134)

one clearly has kρk

∗

≤ kρk

B,ρ

0

for all ρ ∈

[[A

G

]]

QR

. For the converse, one can use the

same technique as for lemma 16, to prove that

kρk

B,ρ

0

≤ kρk

∗

for the case of a local state

ρ ∈ [[A

G

]]

(B)

ρ

0

LR

. Thus kρk

∗

= kρk

B,ρ

0

for local

states ρ ∈ [[A

G

]]

(B)

ρ

0

LR

. It is then straightforward

to conclude that equality holds for every quasi

local state ρ ∈ [[A

G

]]

(B)

ρ

0

QR

.

Notice that two partitions B and B

0

along with

two local reference states ρ

0

and ρ

0

might deﬁne

the same space [[A

G

]]

(B)

ρ

0

QR

≡ [[A

G

]]

(B

0

)

ρ

0

QR

. This is

the case if for every σ ∈ [[A

G

]]

(B)

ρ

0

QR

there exists a

state σ

0

∈ [[A

G

]]

(B

0

)

ρ

0

QR

such that, σ = σ

0

, as func-

tionals on [[

¯

A

G

]]

QR

. We then provide the following

deﬁnition.

Deﬁnition 67. We say that the two reference

local states (B, ρ

0

) and (B

0

, ρ

0

) are compatible,

and denote this relation by the symbol (B, ρ

0

) ∼

(B

0

, ρ

0

), if for all σ ∈ [[A

G

]]

(B)

ρ

0

QR

there exists a

state σ

0

∈ [[A

G

]]

(B

0

)

ρ

0

QR

such that, for every a ∈

[[

¯

A

G

]]

QR

, one has

(a|σ) = (a|σ

0

).

The equivalence classes of pairs (B, ρ

0

) under

the above relation will be denoted by [(B, ρ

0

)].

The norm k·k

B,ρ

0

can be extended to the space

of ﬁnite linear combinations of generalised quasi-

local states.

Deﬁnition 68. Let σ

i

∈ [[A

G

]]

(B

(i)

)

ρ

(i)

0

QR

, for i =

1, . . . , k, with [(B

(i)

, ρ

(i)

0

)] 6= [(B

(j)

, ρ

(j)

0

)]. Then

we deﬁne



k

X

i=1

a

i

σ

i



Q

:

=

k

X

i=1

|a

i

|kσ

i

k

B

(i)

,ρ

(i)

0

.

The space of ﬁnite sums as from deﬁnition 68

is a real vector subspace of [[A

G

]]

R

. Moreover,

as a real vector space, it is equipped with the

norm k·k

Q

. We can deﬁne the space of Cauchy

sequences in this space, and complete it by the

usual procedure. This leads us to the following

deﬁnition of the space of quasi-local states.

Deﬁnition 69. The space of generalised quasi-

local states of A

G

, denoted as [[A

G

]]

QR

, is

[[A

G

]]

QR

:

=

M

[B,ρ

0

]

[[A

G

]]

(B)

ρ

0

QR

,

where the direct sum denotes closure of the space

of ﬁnite linear combinations in the norm k·k

Q

.

The norm introduced above allows us to prove

that [[A

G

]]

QR

is a space of bounded linear func-

tionals on [[

¯

A

G

]]

QR

, namely a subspace of [[A

G

]]

R

.

Lemma 83. Every quasi-local state ρ ∈ [[A

G

]]

QR

is a continuous functional on the space of quasi-

local eﬀects a ∈ [[

¯

A

G

]]

QR

, and

|(a|ρ)| ≤ kak

sup

kρk

Q

(135)

Proof. Let ρ ∈ [[A

G

]]

R

be a ﬁnite sum of the form

ρ =

k

X

i=1

a

i

σ

i

, σ

i

∈ [[A

G

]]

(B

(i)

)

ρ

0

QR

. (136)

Then

(a|ρ)

:

=

k

X

i=1

a

i

(a|σ

i

). (137)

One can easily realise that |(a|ρ)| ≤ kak

sup

kρk

Q

.

Finally, by the above inequality, also the limit

of a Cauchy sequence in [[A

G

]]

QR

is a continuous

linear functional on [[

¯

A

G

]]

QR

, and satisﬁes

|(a|ρ)| ≤ kak

sup

kρk

Q

.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 62

We can now prove the following important re-

sult.

Lemma 84. The space [[A

G

]]

QR

is a closed sub-

space of the extended space of states [[A

G

]]

R

. On

[[A

G

]]

QR

it holds that k·k

Q

≡ k·k

∗

.

Proof. We only need to prove that k·k

∗

≡ k·k

Q

on ﬁnite sums of the form of eq. (136), since we

already proved that [[A

G

]]

QR

is obtained closing

a linear manifold in the space of bounded lin-

ear functionals on [[

¯

A

G

]]

QR

. The proof follows ex-

actly the same argument as for lemma 82, using

eq. (135) instead of eq. (134).

Notice that all local states allow us to de-

ﬁne states of arbitrary ﬁnite regions S ⊆ G, by

Eq. (122). We will now extend the deﬁnition of

local state for all quasi-local states in a ﬁxed sec-

tor [[A]]

(B)

ρ

0

Q

, as follows. First, if ρ is quasi local,

then kρ

n

− ρ

m

k

B,ρ

0

≤ ε, and k(ρ

n

− ρ

m

)|

S

k

op

≤

kρ

n

− ρ

m

k

B,ρ

0

≤ ε. Thus, being the set of states

[[A

S

]] compact, ρ

S

:

= lim

n→∞

ρ

n

|

S

∈ [[A

S

]].

The same is true for ﬁnte linear combinations,

as well as limits of Cauchy sequences thereof. In-

deed, given a ﬁnite combination of states ρ =

P

i

a

i

ρ

i

, one has

ρ

|S

:

=

X

i

a

i

ρ

i|S

∈ [[A

S

]]

R

. (138)

Finally, if τ : N → [[A]]

QR

, and kτ

n

− τ

m

k

Q

≤ ε,

then clearly k(τ

n

− τ

m

)

|

S

k

op

≤ ε, and thus

τ

|S

:

= lim

n→∞

(τ

n

)

|S

∈ [[A

S

]]

R

. (139)

In view of the above considerations, we can now

deﬁne the set of states in [[A

G

]]

QR

.

Deﬁnition 70. The set [[A

G

]]

Q

of quasi-local

states is the (convex) set of generalised states ρ

for which, for every ﬁnite region S ∈ R

(G)

, the

restriction ρ

|S

of ρ to S is a state in [[A

S

]]. The

positive cone [[A

G

]]

Q+

of A

G

is

[[A

G

]]

Q+

:

= {σ ∈ [[A

G

]]

QR

| ∃λ ≥ 0, ρ ∈ [[A

G

]]

Q

: σ = λρ}.

The set of deterministic quasi-local states of A

G

,

denoted as [[A

G

]]

Q1

, is

[[A

G

]]

Q1

:

= {ρ ∈ [[A

G

]]

Q

| (e

G

|ρ) = 1}.

Lemma 85. Deﬁnition 70 is compatible with def-

inition 16

Proof. We need to prove that the deﬁnition of

restriction to the ﬁnite region S provided in

Eq. (33) is equivalent to that of Eq. (139). We

start from the case of local states. In this case,

since eﬀects are separating for states of ﬁnite

dimensional systems, one has (a

S

|ρ

T

) = (a

S

⊗

e

T \S

|ρ

|T

⊗ρ

0|S\T

), and thus ρ

|S

= ρ

|(T ∩S)

⊗ρ

0|S\T

,

which is precisely the same as in Eq. (122). The

statement in the case of a general state ρ ∈

[[A

G

]]

Q

is then straightforwardly proved, remind-

ing that |(a

S

|ρ − ρ

n

)| ≤ ka

S

k

sup

kρ − ρ

n

k

Q

.

Corollary 19. One has the following inclusions

[[A

G

]]

Q

⊆ [[A

G

]], [[A

G

]]

Q1

⊆ [[A

G

]]

1

. (140)

C Proof of identity 94

In order to keep the notation lighter, we do the

calculation for the case of C

∼

=

I. The proof can

then be adapted to the general case by suitably

padding all transformations with I

C

and extend-

ing the region

˜

S to

˜

SC.

(V A V

−1

⊗ I

H

1

)

†

a

˜

S

= W

−1†

(A ⊗ I

H

1

)

†

W

†

a

˜

S

=W

−1†

(A ⊗ I

H

1

)

†

S

†

N

+∗

S

∪N

+

S

0†

N

+∗

S

∪S

a

˜

S

=S

0†

P

+∗

S

∪N

+∗

S

∪R

S

†

N

+

S

∪N

+∗

S

∪R

(A ⊗ I

H

1

)

†

× S

†

N

+∗

S

∪N

+

S

0†

N

+∗

S

∪S

a

˜

S

=S

0†

P

+∗

S

∪N

+∗

S

∪R

S

†

N

+

S

∪N

+∗

S

∪R

(A ⊗ I

H

1

)

†

× S

†

N

+

S

∪N

+∗

S

∪R

S

0†

P

+∗

S

∪N

+∗

S

∪R

a

˜

S

=S

0†

P

+∗

S

∪N

+∗

S

∪R

S

†

R

(A ⊗ I

H

1

)

†

S

†

R

S

0†

P

+∗

S

∪N

+∗

S

∪R

a

˜

S

=S

0†

P

+∗

S

∪N

+∗

S

∪R

(I

H

0

⊗ A )

†

S

0†

P

+∗

S

∪N

+∗

S

∪R

a

˜

S

=S

0†

R

(I

H

0

⊗ A )

†

S

0†

R

[a

˜

S

]

=S

0†

R

S

†

R

(A ⊗ I

H

0

)

†

S

†

R

S

0†

R

[a

˜

S

],

where in the third identity we used Eq. (91), ob-

serving that (A ⊗ I

H

1

)

†

S

†

N

+∗

S

∪N

+

S

0†

N

+∗

S

∪S

[a

˜

S

] ∈

[[

¯

A

(G)

(N

+∗

S

∪R,0

∪ (N

+

S

, 1)]]

QR

. In the fourth identity

we used the fact that a

˜

S

can be padded with e

X

to obtain an eﬀect ˜a

X

on the region

˜

S ∪ X, and

then W

†

˜a

X

is an eﬀect equivalent to W

†

a

S

. In

the ﬁfth identity we used the fact that S

X

com-

mutes with Y

Y

⊗ I

H

1

for Y ∩ X = ∅, and in

the seventh identity we used the fact that S

0

X

commutes with I

H

0

⊗ Y

Y

for Y ∩ X = ∅.

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 63

D Homogeneity and causal inﬂuence

This section is devoted to the proof that, for π ∈

Π

V

,

π(g)  π(g

0

) ⇔ g  g

0

By deﬁnition, f  f

0

if there exists R 3 f and

∃F ∈ [[A

(G)

f

C → A

(G)

f

C]]

QR

,

(V ⊗ I

C

)F (V

−1

⊗ I

C

) 6∈ [[A

(G)

¯

f

0

C → A

(G)

¯

f

0

C]]

QR

.

(141)

Let now f = π(g) for some π ∈ Π

V

. Now, by

homogeneity, and in particular by Eq. (104), one

has that F

0

∈ [[A

(G)

π(g)

C → A

(G)

π(g)

C]] if and only

if there exists F ∈ [[A

(G)

g

C → A

(G)

g

C]] such that

F

0

= (T

π

⊗ I

C

)F (T

−1

π

⊗ I

C

). Then one has

(V ⊗ I

C

)F

0

(V

−1

⊗ I

C

)

= (V T

π

⊗ I

C

)F (T

−1

π

V

−1

⊗ I

C

)

= (T

π

V ⊗ I

C

)F (V

−1

T

−1

π

⊗ I

C

).

(142)

As a consequence, again by eq. (104), we have

that (V ⊗ I

C

)F

0

(V

−1

⊗ I

C

) ∈ [[A

(G)

¯

f

0

C →

A

(G)

¯

f

0

C]], for f

0

= π(g

0

), if and only if (V ⊗

I

C

)F (V

−1

⊗ I

C

) ∈ [[A

(G)

¯g

0

C → A

(G)

¯g

0

C]]. Fi-

nally, comparing with Eq. (141), this implies that

π(g)  π(g

0

) if and only if g  g

0

.

E Proof of right and left invertibility of

a locally deﬁned GUR

Here we prove right and left invertibility of V

deﬁned starting from Eq. (109). First, notice that

ψ

P

−

R

S

0

N

−

R

P

−

R

\R

e

R

C

a

N

−

R

N

−

R

e

=

ψ

P

−

R

S

0

N

−

R

P

−

R

\R

e

R R

e

C

A

C

e

N

−

R

N

−

R

e

=

ψ

P

−

R

S

0

N

−

R

P

−

R

\R

S

0

N

−

R

P

−

R

e

R R

C

A

C

e

N

−

R

N

−

R

N

−

R

e

=

ψ

P

−

R

e

C

A

0

C

e

N

−

R

N

−

R

e

,

where we used Eq. (94).

Let us now deﬁne W by W

†

b

SC

:

= {(W

†

S

⊗

I

†

C

)b}

N

+

S

C

, where

N

+

S

W

S

:

=

N

+

S

0

S

N

+

S

e

η

S S

.

(143)

Then one can calculate W

†

V

†

a

RC

by the follow-

ing diagram

P

−

R

\R

S

0

N

−

R

P

−

R

e

ψ

P

−

R

S

0

N

−

R

P

−

R

\R

e

R R

a

C

η

N

−

R

N

−

R

N

−

R

e

=

P

−

R

\R

S

0

N

−

R

P

−

R

e

R C

A

0

C

e

η

N

−

R

N

−

R

N

−

R

e

=

P

−

R

\R

S

0

N

−

R

P

−

R

S

0

N

−

R

P

−

R

\R

e

R R

A

R

e

C C

e

η

N

−

R

N

−

R

N

−

R

e

=

P

−

R

\R

e

R

a

C

.

Finally, we observe that [(a ⊗ e

P

−

R

\R

)

P

−

R

C

] =

[a

RC

], then W

†

V

†

= I

†

G

. Similarly, one can cal-

culate V

†

W

†

b

SC

. First we observe that [b

SC

] =

[(b ⊗ e

P

+

S

\S

)

P

+

S

C

]. Then W

†

b

SC

= {(W

†

P

+

S

(b ⊗

e

N

+

P

+

S

\N

+

S

)}

N

+

P

+

S

C

, and

N

+

P

+

S

\N

+

S

0

P

+

S

N

+

P

+

S

e

N

+

S

P

+

S

\S

e

C

b

η

P

+

S S

=

N

+

P

+

S

\N

+

S

0

P

+

S

N

+

P

+

S

e

N

+

S

P

+

S

\S

e

C

B

C

e

η

P

+

S S S

e

=

N

+

P

+

S

\N

+

S

0

P

+

S

N

+

P

+

S

0

P

+

S

N

+

P

+

S

e

N

+

S

P

+

S

\S

C

B

C

e

P

+

S

e

η

P

+

S S S

P

+

S

e

=

N

+

P

+

S

\N

+

S

e

N

+

S

B

00

N

+

S

e

C C

e

η

P

+

S

e

,

where we used Eq. (97). Now, for V

†

W

†

[b

SC

] we

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 64

have

ψ

P

−

N

+

P

+

R

S

0

P

+

P

+

R

N

+

P

+

R

S

0

P

+

R

N

+

P

+

R

e

C

b

P

−

N

+

P

+

R

\N

+

P

+

R

e

R

P

+

P

+

R

P

+

P

+

R

e

η

P

+

R

P

+

R

\R

e

=

ψ

P

−

N

+

P

+

R

S

0

P

+

P

+

R

N

+

P

+

R

\N

+

R

e

N

+

R

B

00

N

+

R

e

C C

e

P

−

N

+

P

+

R

\N

+

P

+

R

e

P

+

P

+

R

P

+

P

+

R

e

=

C

b

R

P

+

P

+

R

\R

e

,

where Eq. (97) was used in the last step. The

arguments are straightforwardly generalised to

b ∈ [[

¯

A

(G)

S

¯

C]]

QR

.

F Proof of theorem 10

Proof. In the following, G

i

:

= (G, i) for i = 0, 1.

The argument in appendix E shows that

G

0

W

G

0

e

G

1

G

1

=

G

0

e

G

1

V

−1

G

0

,

where (G, A, V

−1†

) is a GUR. Moreover, since

G

0

S

G

1

G

1

G

0

e

=

G

0

e

G

1

,

one has

G

0

e

G

1

=

G

0

W

−1

G

0

W

G

0

e

G

1

G

1

G

1

=

G

0

S

G

1

W

G

1

S

G

0

e

G

1

G

0

G

0

G

1

V

−1

G

1

=

G

0

S

G

1

W

G

1

V

−1

G

0

G

1

G

0

G

0

e

,

namely

G

1

G

0

e

=

G

1

W

G

1

V

−1

G

0

G

0

G

0

e

.

On the other hand, one can write the latter as

G

1

V

G

1

G

0

e

=

G

1

W

G

1

G

0

G

0

e

, (144)

or equivalently

G

0

e

G

1

V

G

1

=

G

0

W

−1

G

0

e

G

1

G

1

.

Now, let us consider

G

0

˜

S

G

1

e

G

1

G

0

G

2

:

=

G

0

S

G

1

e

G

1

W

−1

G

1

G

0

W

G

0

G

2

G

2

G

2

=

G

0

W

G

0

G

2

W

G

2

G

2

G

1

G

1

e

=

G

0

W

G

0

G

2

V

G

2

G

2

G

1

e

,

from which one concludes that

G

0

˜

S

G

1

e

G

1

G

0

=

G

1

e

G

0

X

G

0

,

for some X . Thus

G

0

W

G

0

G

2

V

G

2

G

2

=

G

0

X

G

0

G

2

.

Finally, this implies that W = X ⊗ V

−1

, and

in particular, by Eq. (144), it must be X = V ,

namely W = V ⊗ V

−1

. Consequently, we have

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 65

that

C

V

−

L

(A

R

)

C

G

0

G

0

G

1

=

C

A

R

C

G

0

W

−1

G

0

G

0

W

G

0

G

1

G

1

G

1

=

C

A

R

C

G

0

V

−1

G

0

G

0

V

G

0

G

1

,

and

G

0

G

1

V

+

L

(A

R

)

G

1

C C

=

G

0

W

−1

G

0

W

G

0

G

1

G

1

A

R

G

1

G

1

C C

=

G

0

G

1

V

G

1

A

R

G

1

V

−1

G

1

C C

.

G Proof of lemmas 66 and 67

We start with the proof of lemma 66.

Proof. (1) Notice that N

0±

e

H

= φ(S

±

) = φ(N

±

e

G

).

Then |N

0±

e

H

| ≤ |N

±

e

G

|, with strict inequality if and

only if φ(h

a

) = φ(h

b

) for some h

a

6= h

b

∈ N

±

e

G

.

However, this is impossible because it would be

equivalent to φ(h

a

h

−1

b

) ∈ R

0

, namely h

a

= h

b

,

contrarily to the hypothesis. Thus, |N

0±

e

H

| =

|N

±

e

G

|. Invariance of neighbourhoods under left-

multiplication ﬁnally gives the thesis.

(2) Let now f ∈ P

0±

e

H

= N

0∓

N

0±

e

H

, namely there

exists h ∈ N

0±

e

H

such that f ∈ N

0∓

h

, or equiva-

lently h ∈ N

0±

f

. Then, f ∈ P

0±

e

H

iﬀ there exists

h ∈ N

0±

e

H

∩ N

0±

f

. Equivalently, there exists h ∈ H

such that h = φ(h

a

) and h = fφ(h

b

), namely

f = φ(h

a

h

−1

b

). Then P

0±

e

H

= {φ(h

a

h

−1

b

) | h

a

, h

b

∈

N

±

e

G

}. By a similar argument, we have that

P

±

e

G

= {h

a

h

−1

b

| h

a

, h

b

∈ N

±

e

G

}. Consequently,

|P

0±

e

H

| ≤ |{h

a

h

−1

b

| h

a

, h

b

∈ N

+

e

G

}| = |P

±

e

G

|. Now,

strict inequality in the last relation would require

that φ(h

a

h

−1

b

) = φ(h

d

h

−1

c

), contrarily to the hy-

pothesis. We conclude that |P

0±

e

H

| = |P

±

e

G

|. Also

in this case the thesis can be obtained by invari-

ance of neighbourhoods under left-multiplication.

(3) Let N

0±

f

1

∩ N

0±

f

2

= ∅. Then clearly for every

g

i

∈ φ

−1

(f

i

) it must be N

±

g

1

∩ N

±

g

2

= ∅. Let

then now h ∈ N

0±

f

1

∩ N

0±

f

2

. This implies that

h = f

1

φ(h

a

) = f

2

φ(h

b

) for h

a

, h

b

∈ N

±

e

G

. Conse-

quently, f

2

= f

1

φ(h

a

h

−1

b

). Now, either f

1

= f

2

,

and then choosing g

1

= g

2

the thesis trivially fol-

lows, or φ(h

a

h

−1

b

) 6∈ R

0

, and then h

a

h

−1

b

6∈ R.

One can then set g

2

= g

1

h

a

h

−1

b

, and verify that

g

2

∈ φ

−1

(f

2

), as well as k = g

1

h

a

= g

2

h

b

∈

N

±

g

1

∩ N

±

g

2

. Thus, h = φ(k), and consequently

N

0±

f

1

∩ N

0±

f

2

⊆ φ(N

±

g

1

∩ N

±

g

2

). This implies that

|N

0±

f

1

∩N

0±

f

2

| ≤ |N

±

g

1

∩N

±

g

2

|, with a strict inequality

iﬀ φ(k

1

) = φ(k

2

) for k

1

6= k

2

∈ N

±

g

1

∩ N

±

g

2

. How-

ever, form the proof of item (1) we know that if

k

1

6= k

2

∈ N

±

g

it must be φ(k

1

) 6= φ(k

2

). Thus,

|N

0±

f

1

∩ N

0±

f

2

| = |φ(N

±

g

1

∩ N

±

g

2

)|, and N

0±

f

1

∩ N

0±

f

2

=

φ(N

±

g

1

∩ N

±

g

2

).

(4) Let h ∈ P

0∓

f

2

∩ N

0±

f

1

. Then h = f

2

φ(h

−1

c

h

b

) =

f

1

φ(h

a

), with h

a

, h

b

, h

c

∈ N

±

e

G

. This implies

that f

2

= f

1

φ(h

a

h

−1

b

h

c

). We can now set g

2

:

=

g

1

h

a

h

−1

b

h

c

with g

1

∈ φ

−1

(f

1

). In this way,

one has f

2

= φ(g

2

). Let us set k

:

= g

1

h

a

=

g

2

h

−1

c

h

b

∈ P

∓

g

2

∩ N

±

g

1

. Then, h = φ(k), and

P

0∓

f

1

∩N

0±

f

1

⊆ φ(P

∓

g

2

∩N

±

g

1

). By the same argument

as for item (3), the thesis follows.

(5) Let h ∈ P

±

g

1

∩ N

∓

g

2

. Then h = g

1

h

a

h

−1

b

, h =

g

2

h

−1

c

, for some h

a

, h

b

, h

c

∈ N

±

e

G

, and thus g

2

=

g

1

h

a

h

−1

b

h

c

. On the other hand, by hypothesis,

for every h

x

∈ N

±

e

G

one has g

2

6= g

1

h

x

. Collecting

all the informations, we have h

a

h

−1

b

h

c

h

−1

x

6∈ R,

i.e. φ(h

a

h

−1

b

h

c

h

−1

x

) 6∈ R

0

, for every h

x

∈ N

±

e

G

.

Then one has φ(g

2

) = φ(g

1

)φ(h

a

h

−1

b

h

c

). By hy-

pothesis, it is also φ(g

2

) = φ(g

1

)φ(h

d

) for some

h

d

∈ N

±

e

G

. This is clearly absurd, thus it must be

P

±

g

1

∩ N

∓

g

2

= ∅.

Now, we provide the proof of lemma 67

Proof. (1) Let f

2

∈ P

0±

f

1

, namely f

2

=

f

1

φ(h

a

h

−1

b

) for h

a

, h

b

∈ N

±

e

G

. If f

1

= f

2

, it

is suﬃcient to choose g

1

= g

2

∈ φ

−1

(f

1

). Let

now f

1

6= f

2

. Then, let g

1

∈ φ

−1

(f

1

), and

g

2

:

= g

1

h

a

h

−1

b

. By construction, φ(g

2

) = f

2

and

g

2

∈ P

±

g

1

. Now, let h ∈ P

0±

f

1

∩ P

0±

f

2

, namely h =

f

1

φ(h

f

h

−1

e

) = f

2

φ(h

c

h

−1

d

) = f

1

φ(h

a

h

−1

b

h

c

h

−1

d

).

Then we have φ(h

a

h

−1

b

h

c

h

−1

d

h

e

h

−1

f

) ∈ R

0

, and

necessarily h

a

h

−1

b

h

c

h

−1

d

h

e

h

−1

f

∈ R. This implies

that g

1

h

f

h

−1

e

= g

1

h

a

h

−1

b

h

c

h

−1

d

= g

2

h

c

h

−1

d

. Now,

setting k

:

= g

1

h

f

h

−1

e

, we have k = g

2

h

c

h

−1

d

∈

P

±

g

1

∩ P

±

g

2

, and h = φ(k). Thus, P

0±

f

1

∩ P

0±

f

2

⊆

Accepted in Quantum 2020-07-03, click title to verify. Published under CC-BY 4.0. 66

φ(P

±

g

1

∩ P

±

g

2

). Finally, by the same argument as

for the proof of item (2) of lemma 66, we get the

thesis.

(2) The hypothesis clearly implies that N

±

g

1

∩

N

±

g

2

= ∅. Suppose now that h ∈ P

±

g

1

∩ P

±

g

2

. Then

h = g

1

h

a

h

−1

b

= g

2

h

d

h

−1

c

for h

a

, h

b

, h

c

, h

d

∈ N

±

e

G

.

This implies that g

2

= g

1

h

a

h

−1

b

h

c

h

−1

d

, and con-

sequently φ(g

2

) = φ(g

1

)φ(h

a

h

−1

b

h

c

h

−1

d

). On the

other hand, we proved that for every h

e

, h

f

∈

N

±

e

G

one has g

2

6= g

1

h

f

h

−1

e

, while by hypothesis

φ(g

2

) = φ(g

1

)φ(h

f

h

−1

e

) for some h

e

, h

f

∈ N

±

e

G

.

Thus, while h

a

h

−1

b

h

c

h

−1

d

h

e

h

−1

f

6∈ R, it must be

φ(h

a

h

−1

b

h

c

h

−1

d

h

e

h

−1

f

) 6∈ R

0

. We reached a con-

tradiction, and thus it must be P

±

g

1

∩P

±

g

2

= ∅.

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